On Homological Stability for Configuration Spaces

on Closed Background Manifolds

Federico Cantero, Martin Palmer^{1}

Received: October 5, 2014 Revised: April 8, 2015

Communicated by Stefan Schwede

Abstract. We introduce a new map between configuration spaces of points in a background manifold – the replication map – and prove that it is a homology isomorphism in a range with certain coefficients. This is particularly of interest when the background manifold is closed, in which case the classical stabilisation map does not exist. We then establish conditions on the manifold and on the coefficients under which homological stability holds for configuration spaces on closed manifolds. These conditions are sharp when the background manifold is a two-dimensional sphere, the classical counterexample in the field. For field coefficients this extends results of Church [Church, 2012] and Randal-Williams [Randal-Williams, 2013a] to the case of odd characteristic, and for p-local coefficients it improves results of Bendersky– Miller [Bendersky and Miller, 2014].

2010 Mathematics Subject Classification: 55R80, 55P60, 55R25 Keywords and Phrases: Homological stability, configuration spaces, replication map, scanning map, closed background manifolds.

1. Introduction

Let M be a smooth, connected manifold without boundary of dimension n, and with Euler characteristic χ, and denote by C_{k}(M) the unordered configuration space of k points in M:

C_{k}(M) := {q ⊂ M | |q| = k},

which is topologised as a quotient space of a subspace of M^{n}. After removing a point ∗ from M one can define a map

C_{k}(M ∖ {∗}) ⟶ C_{k+1}(M ∖ {∗}),

1Both authors were funded by Michael Weiss’ Humboldt professor grant. The first author was partially supported by the Spanish Ministry of Economy and Competitiveness under grants MTM2010-15831 and MTM2013-42178-P.

called the stabilisation map, which expands the configuration away from ∗ and adds a new point near to it. More generally, one can define such a stabilisation map C_{k}(M) → C_{k+1}(M) using any properly embedded ray in M to bring in a point from infinity (such a ray exists if and only if M is non-compact). Let us assume from now on that the manifold is endowed with a Riemannian metric with injectivity radius bounded below by δ > 0. Define C_{k}^{δ}(M) ⊂ C_{k}(M) × (0, δ) to be the space of pairs (q, ϵ), where q is a configuration whose points are pairwise at distance at least 2ϵ. The projection to C_{k}(M) is a fibre bundle with contractible fibres, hence a homotopy equivalence. The main theorem in [McDuff, 1975] concerns the scanning map

𝒮 : C_{k}^{δ}(M) ⟶ Γ_{c}(Ṫ M)_{k}

which takes values in the space of degree-k compactly-supported sections of the fibrewise one-point compactification of T M (see §3.1).

Definition 1.1 Given a properly embedded ray in M and an abelian group A, define the function μ = μ[M]: ℕ → ℕ to be the pointwise maximum f: ℕ → ℕ such that the stabilisation map C_{k}(M) → C_{k+1}(M) induces isomorphisms on H_{∗}(−; A) in the range ∗ ⩽ f(k). This is called the stable range of the stabilisation map. Given a Riemannian metric on M with injectivity radius bounded below by δ > 0 and an abelian group A, the function ν = ν[M]: ℕ → ℕ is defined to be the pointwise maximum f: ℕ → ℕ such that the scanning map 𝒮 : C_{k}^{δ}(M) → Γ_{c}(Ṫ M)_{k} induces isomorphisms on H_{∗}(−; A) in the range ∗ ⩽ f(k). This is called the stable range of the scanning map.

Henceforth the term “stable range” will by default refer to the stable range ν of the scanning map.

Theorem ([McDuff, 1975]) For non-compact M the function μ[M] diverges and

ν[M](k) = min_{j⩾k}{μ[M](j)}.

The inequality ν[M] ⩾ ν[M∖{∗}] holds for all M, so the function ν[M] diverges for all M.

No explicit lower bound for μ[M] was given in [McDuff, 1975], but the following lower bounds have since been proved:

• μ[M](k) ⩾ \frac{k}{2} if A = ℤ and dim(M) ⩾ 2, by [Segal, 1979; Randal-Williams, 2013a].

• μ[M](k) ⩾ k if A = ℚ and either dim(M) ⩾ 3 or M is non-orientable, by [Randal-Williams, 2013a; Knudsen, 2014].

• μ[M](k) ⩾ k − 1 if A = ℚ and M is orientable, by [Church, 2012; Knudsen, 2014].

• μ[M](k) ⩾ k if A = ℤ[\frac{1}{2}] and dim(M) ⩾ 3, by [Kupers and Miller, 2014b].

See also Propositions A.2 and B.3. Further improvements to the lower bound are possible under extra hypotheses ([Church, 2012, Proposition 4.1] and [Kupers and Miller, 2014b, Remark 4.5]). Some of these results can be also deduced from [Milgram and Löffler, 1988; Bödigheimer et al., 1989; Félix and Thomas, 2000]. McDuff’s theorem says that the homology of configuration spaces C_{k}(M) on a non-compact manifold M stabilises, i.e., is independent of k in a diverging range of degrees. For closed manifolds M stabilisation maps do not exist – this leaves open the question of when the homology of configuration spaces on closed background manifolds stabilises.

Stability for p-torsion. Let Ṫ M denote the fibrewise one-point compactification of the tangent bundle of M and let Γ_{c}(−) denote the space of compactly-supported sections. By the main result in [Møller, 1987], for each k ∈ ℤ the localisation of the path-component Γ_{c}(Ṫ M)_{k} at a prime p is homotopy equivalent to the path-component Γ_{c}(Ṫ M_{(p)})_{k} of the space of compactly-supported sections of the fibrewise localisation of Ṫ M.^{2} In [Bendersky and Miller, 2014] Bendersky and Miller proved the existence of homotopy equivalences

(1.1)  Γ_{c}(Ṫ M_{(p)})_{k} ⟶ Γ_{c}(Ṫ M_{(p)})_{j}

whenever • p ⩾ \frac{n+3}{2} and M is odd-dimensional,

• p ⩾ \frac{n+3}{2} and \frac{2k−χ}{2j−χ} is a unit in ℤ_{(p)},
• Ṫ M is trivial and \frac{2k−χ}{2j−χ} is a unit in ℤ_{(p)}.

Using McDuff’s theorem one obtains a zigzag of ℤ_{(p)}-homology isomorphisms in the stable range:

(1.2)  C_{k}(M) ⟶ Γ_{c}(Ṫ M)_{k} ⟶ Γ_{c}(Ṫ M)_{j} ⟵ C_{j}(M).

We will show that linearly independent pairs of sections of T M ⊕ ϵ give rise to families of fibrewise homotopy equivalences of Ṫ M after localisation, and hence maps as in (1.1) for certain k and j, from which we are able to extend the results of Bendersky and Miller to all odd primes and under certain conditions to the prime 2. For a number k ∈ ℤ, we denote by (k)_{p} the p-adic valuation of k, and observe that \frac{j}{k} is a unit in ℤ_{(p)} if and only if (k)_{p} = (j)_{p}. If ℓ is a collection of primes, the ℓ-adic valuation is the sequence of all p-adic valuations with p ∈ ℓ.

Theorem A Let M be a closed, connected, smooth manifold. If M is odddimensional, there are zigzags of maps as in (1.2) inducing isomorphisms in the stable range:

2To ensure that the localisation of Γ_{c}(Ṫ M_{(p)})_{k} exists, we need to assume here that M has the homotopy type of a finite complex. However, for the purpose of proving homological stability results, we may assume this without loss of generality; see §2.1.

(1.3)  H_{∗}(C_{k}(M); ℤ) ≅ H_{∗}(C_{k+1}(M); ℤ) if dim M = 3, 7
(1.4)  H_{∗}(C_{k}(M); ℤ) ≅ H_{∗}(C_{k+2}(M); ℤ)
(1.5)  H_{∗}(C_{k}(M); ℤ[\frac{1}{2}]) ≅ H_{∗}(C_{k+1}(M); ℤ[\frac{1}{2}]).

If M is even-dimensional with Euler characteristic χ, then for each set ℓ of primes (assuming 2 ∉ ℓ if χ is odd) there are zigzags of maps as in (1.2) inducing isomorphisms in the stable range:

(1.6)  H_{∗}(C_{k}(M); ℤ_{(ℓ)}) ≅ H_{∗}(C_{j}(M); ℤ_{(ℓ)}) if (2k − χ)_{ℓ} = (2j − χ)_{ℓ}.

In particular there are integral homology isomorphisms between C_{k}(M) and C_{χ−k}(M) in the stable range.

Observe that since these isomorphisms are induced by zigzags of maps, they also give isomorphisms between the cohomology rings of configuration spaces. In Proposition 2.10 we show that when M is an odd-dimensional sphere, this method cannot be used to improve Theorem A. In Proposition 2.12 we prove that our theorem is sharp when M is an even-dimensional sphere: if n is even and k is in the stable range with respect to homological degree n − 1, then

(1.7)  H_{n−1}(C_{k}(S^{n}); ℤ) ≅ τH_{n−1}(Ω_{0}^{n}S^{n}; ℤ) ⊕ ℤ/(2k − 2),

where τG is the torsion of G. In particular, if j is also in the stable range:

H_{n−1}(C_{k}(S^{n}); ℤ_{(ℓ)}) ≅ H_{n−1}(C_{j}(S^{n}); ℤ_{(ℓ)}) ⇔ (2k − χ)_{ℓ} = (2j − χ)_{ℓ}.

This generalises the computation of H_{1}(C_{k}(S^{2}); ℤ) (which follows from the presentation of π_{1}(C_{k}(S^{2})) given by [Fadell and Van Buskirk, 1962]).

Replication maps. Our next result involves a new map between configuration spaces, defined whenever M admits a non-vanishing vector field, which induces some of the homology isomorphisms of Theorem A. This map (or rather its effect on π_{1}) has been considered before in the case M = ℝ^{2} in the context of the Burau representations of the classical braid groups [Blanchet and Marin, 2007]. It has also appeared in §7 of [Martin and Woodcock, 2003]. However, to our knowledge its homological stability properties have not previously been studied. A homomorphism π_{1}(C_{k}(M)) → π_{1}(C_{k+1}(M)) (which is not induced by a map of spaces) was defined using a similar idea in [Berrick et al., 2006] (see page 283), where it was used to show that the collection {π_{1}(C_{k}(M))} is a crossed simplicial group when M admits a non-vanishing vector field. This map is especially interesting when M is closed, in which case it allows one to compare configuration spaces which do not admit any stabilisation map. It is also useful when M is open: we will use this map in the case of open manifolds to prove Theorem D, which concerns closed manifolds. Let v be a non-vanishing vector field on M of norm 1. Define the r-replication map ρ_{r} = ρ_{r}[v]: C_{k}^{δ}(M) → C_{rk}^{δ}(M) by adding r − 1 points near each point of the configuration in the direction of the vector field v:

ρ_{r}[v](q = \{q_{1}, …, q_{k}\}, ϵ) = (\{exp(\frac{jϵ}{r}v(q_{i})) | \begin{matrix}i=1,...,k \\ j=0,...,r−1 \end{matrix} \}, \frac{ε}{2r} ).

Theorem B Let r ⩾ 2. If M admits a non-vanishing vector field v and ℓ is a set of primes each not dividing r, then the homomorphism induced by ρ_{r}[v]:

H_{∗}(C_{k}^{δ}(M); ℤ_{(ℓ)}) ⟶ H_{∗}(C_{rk}^{δ}(M); ℤ_{(ℓ)})

is an isomorphism in the stable range. If M is not closed, then it is always injective.

Remark 1.2 Observe that the map ρ_{r} does not induce isomorphisms on rtorsion in general. For example take M to be simply-connected and of dimension at least 3. Then π_{1}(C_{k}(M)) ≅ Σ_{k} and H_{1}(C_{k}(M)) ≅ ℤ/2, given by the sign of the permutation. The map Σ_{k} → Σ_{2k} induced by ρ_{2} on π_{1} sends a permutation σ to the concatenation (σ, σ), whose sign is the square of the sign of σ, therefore zero. Hence the map induced on first homology by ρ_{2} is zero. In particular this shows that ρ_{2} cannot be homotopic to a composition of stabilisation maps.

Configurations with labels and the intrinsic replication map. Given a fibre bundle θ: E → M with path-connected fibres, one can define the configuration space C_{k}(M; θ) with labels in θ by

C_{k}(M; θ) = {{q_{1}, …, q_{k}} ⊂ E | θ(q_{i}) ≠ θ(q_{j}) for i ≠ j}.

Configuration spaces with labels admit stabilisation maps, scanning maps and replication maps (see [Kupers and Miller, 2014b] and Definition B.1 in this article for the stabilisation map, and Section 4 for the other two maps) which induce homology isomorphisms in a range, which we call the stable range with labels in θ. To define the replication and the scanning map it is more convenient to use the following alternative model:

C_{k}^{δ}(M; θ) = {(q, ϵ, s) | (q, ϵ) ∈ C_{k}^{δ}(M), s: B_{ϵ/2}(q) → E a section of θ},

where B_{ϵ/2}(q) means the (disjoint) union of the (ϵ/2)-balls around q for each q ∈ q. So a point in this space consists of a configuration q with prescribed pairwise separation, together with a choice of label on a small contractible neighbourhood of each configuration point. If θ: E → M factors through the unit sphere bundle of T M with a map φ: E → S(T M), then it is possible to define a new map which we call the intrinsic replication map � _{r}: C_{k}^{δ}(M; θ) ⟶ C_{rk}^{δ}(M; θ). It sends the labelled configuration (q = {q_{1}, …, q_{k}}, ϵ, s: B_{ϵ/2}(q) → E) to the labelled configuration

(\{exp(\frac{jϵ}{r}φs(q_{i})) | \begin{matrix}i=1,...,k \\ j=0,...,r−1 \end{matrix} \}, \frac{ε}{2r} , restriction of s).

In contrast with the (extrinsic) replication map of Theorem B, this map is defined for every manifold M.

Theorem C Let r ⩾ 2 and let ℓ be a set of primes each not dividing r. Then the map � _{r}: C_{k}^{δ}(M; θ) → C_{rk}^{δ}(M; θ) induces isomorphisms on homology with ℤ_{ℓ}-coefficients in the stable range with labels in θ.

An extension for field coefficients. The homology of configuration spaces with field coefficients is better understood than the torsion of their integral homology. In fact, complete descriptions of the additive structure of H_{∗}(C_{k}(M); 𝔽) were given by [Milgram and Löffler, 1988] when 𝔽 has characteristic 2 and by [Bödigheimer et al., 1989] when either 𝔽 has characteristic 2 or M is odd-dimensional. The rational structure was further studied by [Félix and Thomas, 2000] and more recently by [Knudsen, 2014], who gave a complete description of the rational cohomology ring of C_{k}(M). From their computations, it follows that the homology with field coefficients always stabilises, unless the manifold is even-dimensional and the field has odd characteristic. These results were proven again by [Church, 2012] (in the rational case) and [Randal-Williams, 2013a] (in all cases) using homological stability methods (also improving the known stable ranges).

Theorem [Milgram and Löffler, 1988; Bödigheimer et al., 1989; Félix and Thomas, 2000; Church, 2012; Randal-Williams, 2013a; Knudsen, 2014] Let M be a connected, smooth manifold of dimension n, let 𝔽 be a field of characteristic p and assume that p(n − 1) is even. Then in the stable range we have isomorphisms H_{∗}(C_{k}(M); 𝔽) ≅ H_{∗}(C_{k+1}(M); 𝔽).

The last part (§5) of this article addresses the question of homological stability when p(n − 1) is odd, in other words for even-dimensional (closed) manifolds and with coefficients in fields of odd characteristic. It does not involve section spaces, but rather uses the result of Theorem B in the case of open manifolds M together with an argument similar to that of [Randal-Williams, 2013a, §9]. If M is a closed, connected manifold one can choose a vector field on M which is non-vanishing away from a point ∗ ∈ M. This vector field (suitably normalised) therefore induces an r-replication map for configuration spaces on M ∖ {∗}, which induces isomorphisms on homology with ℤ[\frac{1}{r}] coefficients in the stable range by Theorem B. We can fit C_{k}(M) into a cofibre sequence in which the other two spaces are suspensions of configuration spaces on M∖{∗}. We can then define stabilisation maps on the other two spaces using the r-replication map and the ordinary stabilisation map, which are isomorphisms on homology localised away from r in the stable range. We will therefore have homological stability for C_{k}(M), with field coefficients of characteristic coprime to r, as long as the square formed by this pair of stabilisation maps commutes. In fact it does not commute in general, but the obstruction to commutativity on homology is a single homology class whose divisibility we can calculate. Thus we obtain the following theorem, where

λ(k) = λ[M](k) := min{ν(k), ν(k − 1) + n − 1, μ(rk − i) | i = 2, …, r}.

Here n is the dimension of M and μ = μ[M ∖ {∗}] and ν = ν[M ∖ {∗}].

Theorem D Let M be a closed, connected, even-dimensional smooth manifold. Choose a field 𝔽 of positive characteristic p and let r ⩾ 2 be an integer coprime to p such that p divides (χ − 1)(r − 1). Then there are isomorphisms

H_{∗}(C_{k}(M); 𝔽) ≅ H_{∗}(C_{rk}(M); 𝔽)

in the range ∗ ⩽ min(λ(k), λ(rk)).

See Remark 5.7 for an explanation of how the function λ[M] arises, and the remark that if μ[M ∖ {∗}] is linear with slope ⩽ dim(M) − 1 and r, k ⩾ 2, then λ[M](k) = μ[M ∖ {∗}](k). This theorem also generalises to configuration spaces with labels in a fibre bundle over M with path-connected fibres. See §5.4 for the proof for configuration spaces without labels and §5.6 for a sketch of the generalisation to configuration spaces with labels (Theorem D′).

Remark 1.3 When M is odd-dimensional the conclusion of Theorem D follows directly from Theorem A. Also, we note that our proof in §5.4 also works for fields of characteristic zero: in this case we must asssume that χ = 1, but the proof then becomes simpler since the square (5.9) commutes up to homotopy (not only on homology). Finally, in the case where the fibre bundle over M factors through the unit sphere bundle S(T M) → M, Theorem D′ follows from Theorem C′.

Combining Theorems A and D. Theorem A says that in odd dimensions there are at most two stable integral homologies, depending on the parity of the number of points k. On the other hand, in even dimensions – even when taking homology with ℤ_{(p)} coefficients – there may be infinitely many different stable homologies: one for each possible p-adic valuation of 2k − χ. In fact this is sharp, as the calculation (1.7) shows. However, the situation is simpler when taking 𝔽_{p} coefficients. From the calculation (1.7) we see that, when n is even and k is in the stable range with respect to degree n − 1, we have

H_{n−1}(C_{k}(S^{n}); 𝔽_{p}) ≅ Tor(H_{n−2}(Ω_{0}^{n}S^{n}), 𝔽_{p}) ⊕ (τH_{n−1}(Ω_{0}^{n}S^{n}) ⊗ 𝔽_{p})
⊕ (ℤ/(2k − 2) ⊗ 𝔽_{p}).

Writing d for the dimension of the first two summands on the right-hand side (which is independent of k), it follows that H_{n−1}(C_{k}(S^{n}); 𝔽_{p}) is either d- or (d + 1)-dimensional depending on whether or not p divides 2k − 2, so there are at most two stable 𝔽_{p}-homologies in this special case. One can combine Theorems A and D to prove that this phenomenon holds more generally:

Corollary E Let M be a closed, connected, even-dimensional smooth manifold and let 𝔽 be a field of odd characteristic p. Then there are canonical (additive) isomorphisms

H_{∗}(C_{k}(M); 𝔽) ≅ H_{∗}(C_{j}(M); 𝔽)

under either of the following conditions:

• min{(2k − χ)_{p}, (χ)_{p} + 1} = min{(2j − χ)_{p}, (χ)_{p} + 1},

• χ ≡ 1 mod p,

in the range ∗ ⩽ min(λ(k), λ(j)).

This is proved in §5.7, where we also partially recover the known homological stability results for odd-dimensional manifolds and fields of characteristic 2 or 0 (see Corollary 5.8).

Number of stable homologies. Homological stability (without an explicit range) for configuration spaces with coefficients in a field 𝔽 can be rephrased as the statement that for each degree i, the set {dimH_{i}(C_{j}(M); 𝔽) | j = k, …, ∞} contains only one element once k is sufficiently large, in other words, the number

nsh_{i}(M; 𝔽) := lim_{k→∞}|{dimH_{i}(C_{j}(M); 𝔽)}_{j=k}^{∞}| ∈ {1, 2, 3, …, ∞}

is equal to 1. As mentioned earlier, we have nsh_{i}(M; 𝔽) = 1 whenever either dim(M) is odd or char(𝔽) is even, and we also have the example that nsh_{1}(S^{2}; 𝔽_{p}) = 2. The above corollary can be viewed as proving that whenever M has non-zero Euler characteristic, nsh_{i}(M; 𝔽) is finite and has the explicit upper bound:

nsh_{i}(M; 𝔽) ⩽ (χ)_{p} + 2

where p = char(𝔽) and χ is the Euler characteristic of M. Moreover, we also have

nsh_{i}(M; 𝔽) = 1 when χ ≡ 1 mod p.

In particular this means that when χ(M) = 1 we have nsh_{i}(M; 𝔽) = 1 for any field 𝔽, in other words homological stability holds for the sequence {C_{k}(M)}_{k=1}^{∞} with coefficients in any field.

Homological periodicity. A consequence of Corollary E is that the sequence of homology groups {H_{i}(C_{k}(M); 𝔽)}_{k=1}^{∞} for fixed M, 𝔽 and i is eventually periodic as k → ∞, as long as χ ≠ 0. To see this, note that by Corollary E it is enough to show that

(1.8)  min{(2k − χ)_{p}, (χ)_{p} + 1} = min{(2(k + a) − χ)_{p}, (χ)_{p} + 1}

for some natural number a independent of k. Note that for any two natural numbers x, y we have the inequality (x + y)_{p} ⩾ min{(x)_{p}, (y)_{p}} and a sufficient condition for equality is that (x)_{p} and (y)_{p} are distinct. Considering the cases (2k − χ)_{p} > (χ)_{p} and (2k − χ)_{p} ⩽ (χ)_{p} separately, and applying this fact, one can easily show that the equation (1.8) holds for a = p^{(χ)_{p}+1}. Hence we have:

Corollary F Let M be a closed, connected, even-dimensional smooth manifold with Euler characteristic χ ≠ 0 and let 𝔽 be a field of odd characteristic p. Then the sequence

(1.9)  {H_{i}(C_{k}(M); 𝔽)}_{k=1}^{∞}

is eventually periodic in k as k → ∞, with period equal to p^{e_{i}(M;𝔽)} for some number e_{i}(M; 𝔽) ⩽ (χ)_{p} + 1. Equivalently, there are (additive) isomorphisms

H_{i}(C_{k}(M); 𝔽) ≅ H_{i}(C_{k+p^{χ(M)p+1}}(M); 𝔽)

for k ≫ i (precisely, in the range i ⩽ λ(k)).

This is similar to a result of Nagpal [Nagpal, 2015, Theorem F], who also proves that the sequence (1.9) is eventually periodic in k as k → ∞ and obtains an explicit period of p^{(i+3)(2i+2)}. The difference is that his result also holds when χ = 0, but on the other hand he assumes that M is orientable, and his upper bound on the period depends on the homological degree. Note however that Corollary E is much stronger than homological periodicity: it implies that the number of stable homologies nsh_{i}(M; 𝔽) is bounded above by (χ)_{p} + 2, whereas Corollary F alone only implies an upper bound of p^{χ(M)_{p}+1}.

Acknowledgements. We thank Oscar Randal-Williams for careful reading of an earlier draft of this paper and for enlightening discussions. The paper has also benefited from conversations with Frederick Cohen, Mark Grant, Fabian Hebestreit, Alexander Kupers and Jeremy Miller. We would also like to thank the anonymous referee for helpful comments and corrections.

2. Homological stability via the scanning map

2.1. Sphere bundles, localisation and fibrewise homotopy equivalences. Let M be a connected manifold and E → M a rank n inner product vector bundle. Let Ė be the fibrewise one point compactification of E. The topological bundle Ė is isomorphic to the unit sphere bundle S(E ⊕ ϵ) of the Whitney sum of E and a trivial line bundle. We denote by ∞ the point at infinity in each fibre. We denote by ι the section with value ∞ and by z the zero section. During the next paragraph we assume temporarily that M is a compact manifold with boundary. Let Γ_{∂}(Ė) ⊂ Γ(Ė) be the subspace of those sections that take value ∞ on the boundary of M. Since the fibre of Ė → M is nilpotent and the pair (M, ∂M) has finitely many non-zero homology groups, then by [Møller, 1987, Theorem 4.1], each connected component of Γ_{∂}(Ė) is also nilpotent. We may therefore consider, for each set of primes ℓ, the localisation Γ_{∂}(Ė)_{(ℓ)}. We may also consider the fibrewise localisation Ė → Ė_{(ℓ)}, and [Møller, 1987, Theorem 5.3] implies that the induced map Γ_{∂}(Ė) → Γ_{∂}(Ė_{(ℓ)}) is a localisation in each component, since (M, ∂M) is a finite relative complex. If M is an arbitrary manifold, we can write it as a union M = ⋃ M^{i} of compact codimension-0 submanifolds with boundary. Let Ė^{i} denote the restriction of Ė to the submanifold M^{i}. The map Γ_{c}(Ė) → Γ_{c}(Ė_{(ℓ)}), induced by the fibrewise localisation Ė → Ė_{(ℓ)}, is then the colimit of the maps Γ_{∂}(Ė^{i}) → Γ_{∂}(Ė_{(ℓ)}^{i}):

\xymatrix{Γ_{∂}(Ė^{i}) \ar[r] \ar[d] & Γ_{∂}(Ė^{i+1}) \ar[r] \ar[d] & ⋯ \ar[r] & Γ_{c}(Ė) \ar[d] \\ Γ_{∂}(Ė_{(ℓ)}^{i}) \ar[r] & Γ_{∂}(Ė_{(ℓ)}^{i+1}) \ar[r] & ⋯ \ar[r] & Γ_{c}(Ė_{(ℓ)}).}

Since the vertical maps induce (componentwise) isomorphisms on homology with ℤ_{(ℓ)}-coefficients, so does their colimit. A bundle endomorphism f of Ė_{(ℓ)} is compactly supported if f ∘ ι = ι outside a compact subset of M. We denote by End_{c}^{r}(Ė_{(ℓ)}) the space of compactly supported endomorphisms which induce on fibres maps of degree r. We denote by end^{r}(Ė_{(ℓ)}) the bundle of pairs (x, f_{x}), where x ∈ M and f_{x}: (Ė_{(ℓ)})_{x} → (Ė_{(ℓ)})_{x} is a map of degree r. By definition End_{c}^{r}(Ė_{(ℓ)}) = Γ_{c}(end^{r}(Ė_{(ℓ)})). By Theorem 3.3 of [Dold, 1963], if r is a unit in ℤ_{(ℓ)}, then any endomorphism in End_{c}^{r}(Ė_{(ℓ)}) admits a fibrewise homotopy inverse. Postcomposition with it induces a homotopy equivalence between path-components

Γ_{c}(Ė_{(ℓ)})_{k} ⟶ Γ_{c}(Ė_{(ℓ)})_{[f](k)},

where [f] denotes the map induced by f on π_{0}Γ_{c}(Ė_{(ℓ)}). We summarize the discussion so far in the following lemma:

Lemma 2.1 If r is a unit in ℤ_{(ℓ)}, f ∈ End_{c}^{r}(Ė_{(ℓ)}) and [f](k) is an integer, then the zigzag

Γ_{c}(Ė)_{k} ⟶ Γ_{c}(Ė_{(ℓ)})_{k} ⟶ Γ_{c}(Ė_{(ℓ)})_{[f](k)} ⟵ Γ_{c}(Ė)_{[f](k)},

where the middle map is given by post-composition with f, induces an isomorphism on homology with ℤ_{(ℓ)}-coefficients.

Remark 2.2 Note that if ℓ = ∅, then (−)_{(ℓ)} is rationalisation (also denoted (−)_{(0)}), whereas if ℓ = Spec ℤ, the set of all primes, this localisation is the identity, i.e., we are not localising at all.

2.2. The degree of a section. Let β be a compactly supported section of π: Ṫ M → M, and let Th(β) be the Thom class in H^{n}(Ṫ M; π^{∗}O), where O is the orientation sheaf of M. The β-degree of a compactly supported section α is

deg_{β}(α) = α^{∗}(Th(β))^{∨} ∈ H_{0}(M; ℤ),

the Poincare dual in M of α^{∗}Th(β) ∈ H_{c}^{n}(M; O). If M is orientable, then Th(β) is the Poincare dual of β_{∗}[M] ∈ H_{n}(Ṫ M; ℤ), and deg_{β}(α) is also equal to the intersection product of α_{∗}[M] and β_{∗}[M] in Ṫ M. We will write deg for deg_{z}, where z is the zero section of Ṫ M. Assume now that M is closed and orientable. The Gysin sequence for the sphere bundle S^{n} →^{i} Ṫ M →^{π} M splits an exact sequence

(2.1)  0 ⟶ H_{n}(S^{n}; ℤ) ⟶^{i_{∗}} H_{n}(Ṫ M; ℤ) ⟶^{π_{∗}} H_{n}(M; ℤ) ⟶ 0.

The zero section z: M → Ṫ M is an inverse of π, so the group H_{n}(Ṫ M) ≅ ℤ⊕ℤ is generated by i_{∗}[S^{n}] and z_{∗}[M]. The fibres over two different points give two disjoint representatives of i_{∗}[S^{n}], therefore i_{∗}[S^{n}] ∩ i_{∗}[S^{n}] = 0. On the other hand, the intersection of the zero section with itself is the Euler characteristic χ of M. And it is also clear that the intersection of i_{∗}[S^{n}] and z_{∗}[M] consists of a single point. The intersection products of 4k (resp. 4k + 2) dimensional manifolds are symmetric (antisymmetric). Therefore we have:

Lemma 2.3 If M is connected, closed, orientable and of dimension n, then the intersection pairing of Ṫ M with respect to the above basis is given by

(\begin{matrix}0 & 1 \\ (−1)^{n} & χ \end{matrix}) .

If α is a section of π, then α_{∗}[M] = (deg(α) − χ, 1) in this basis.

Proof. For the second claim, observe that α is an inverse of π too, so the second component of α_{∗}[M] is the same as the second component of z_{∗}[M]. The first component is obtained from the following equation:

deg(α) = α_{∗}[M] ∩ z_{∗}[M] = (a, 1) (\begin{matrix}0 & 1 \\ (−1)^{n} & χ \end{matrix}) (\begin{matrix}0 \\ 1 \end{matrix}) = a + χ.  □

The Gysin sequence (2.1) applied to the localised bundle S_{(ℓ)}^{n} → Ṫ M_{(ℓ)} → M shows that H_{n}(Ṫ M_{(ℓ)}) ≅ ℤ_{(ℓ)}⊕ℤ. The first factor is generated as a ℤ_{(ℓ)}-module by the fundamental class of the fibre, and the second factor is generated by the image of the fundamental class of M under the zero section. The following definition extends the notion of degree to sections of fibrewise localised sphere bundles.

Definition 2.4 The degree of a section α of Ṫ M_{(ℓ)}, denoted deg(α), is the value a + χ ∈ ℤ_{(ℓ)}, where (a, 1) = α_{∗}[M] in our preferred basis.

2.3. Fibrewise homotopy equivalences of many degrees. Let V_{2}(E⊕ϵ) be the fibrewise Stiefel manifold of E ⊕ ϵ. If σ is a section of Γ(V_{2}(E ⊕ ϵ)_{(ℓ)}) we denote by σ_{0} the image of σ under the localisation of the map that forgets the second vector:

φ_{(ℓ)} : Γ(V_{2}(E ⊕ ϵ)_{(ℓ)}) ⟶ Γ(S(E ⊕ ϵ)_{(ℓ)}).

We denote by Γ_{c}(V_{2}(E ⊕ϵ)_{(ℓ)}) the space of sections σ such that σ_{0} is compactly supported.

Lemma 2.5 Let E be a real inner product bundle over a manifold M. There is a bundle map

V_{2}(E ⊕ ϵ) ⟶ end^{r}(Ė)

for each r ∈ ℤ and therefore, for each set of primes ℓ, there are maps

Φ_{r}^{ℓ}: Γ_{c}(V_{2}(E ⊕ ϵ)_{(ℓ)}) ⟶ End_{c}^{r}(Ė_{(ℓ)})

which are natural with respect to pullback of bundles. If M is closed and E = T M, then Φ_{r}^{ℓ}(σ) sends sections of degree k to sections of degree r(k−deg(σ_{0}))+ deg(σ_{0}).

Proof. A 2-frame in V_{2}(E ⊕ϵ) determines a linear embedding ℝ^{2} → (E ⊕ϵ)_{x}. If we denote by V its orthogonal complement, we obtain canonical isomorphisms ℝ^{2} ⊕ V ≅ (E ⊕ ϵ)_{x} which induce canonical isomorphisms S^{1} ∗ S(V) ≅ S(E ⊕ ϵ). This allows to define a degree r map

\xymatrix{S(E ⊕ ϵ)_{x} ≅ S^{1} ∗ S(V) \ar[r]^{e^{2πir}∗Id} & S^{1} ∗ S(V) ≅ S(E ⊕ ϵ)_{x}.}

After fibrewise localizing and taking sections, one obtains the second map. Observe that the above map fixes the first vector in the 2-frame, hence the image of a section in Γ_{c}(V_{2}(E ⊕ ϵ)_{(ℓ)}) will fix the section ι outside a compact subset. By construction, f^{∗}(Φ_{r}^{ℓ}(σ)) = Φ_{r}^{ℓ}(f^{∗}(σ)), so these maps are natural. Similarly, observe that

(2.2)  End_{c}^{r}(Ė_{(ℓ)}) × Γ(Ė_{(ℓ)}) ⟶ Γ(Ė_{(ℓ)})

is also natural with respect to pullback of bundles. Now we describe the effect of ϕ_{r} := Φ_{r}^{ℓ}(σ) on components of Γ(Ṫ M_{(ℓ)}) when M is closed. Assume first that M is orientable, in which case Ṫ M is also orientable and Lemma 2.3 applies. First we identify the induced map (ϕ_{r})_{∗}: H_{n}(Ṫ M_{(ℓ)}) → H_{n}(Ṫ M_{(ℓ)}). Since ϕ_{r}(σ_{0}) = σ_{0}, we have

(ϕ_{r})_{∗}(deg(σ_{0}) − χ, 1) = (deg(σ_{0}) − χ, 1).

On the other hand, ϕ_{r} acts on the fibre over a point as a map of degree r, hence

(ϕ_{r})_{∗}(1, 0) = (r, 0).

From this we deduce that (ϕ_{r})_{∗} has the form

(\begin{matrix}r & −(r − 1)(deg(σ_{0}) − χ) \\ 0 & 1 \end{matrix}) ,

hence, for an arbitrary section α, we have that

(deg(ϕ_{r}(α)_{∗}[M]) − χ, 1) = ϕ_{r}(α)_{∗}[M] = (ϕ_{r})_{∗}(α_{∗}[M])
= (r(deg(α) − χ) − (r − 1)(deg(σ_{0}) − χ), 1)

and so deg(ϕ_{r}(α)) = r deg(α)−(r−1) deg(σ_{0}) = r(deg(α)−deg(σ_{0}))+deg(σ_{0}). Assume now that M is non-orientable. We take then the orientation cover f: M̃ → M. If s is a section of Ṫ M_{(ℓ)} and σ is a section of V_{2}(T M ⊕ ϵ)_{(ℓ)}, we can pull back both sections along f to obtain a section f^{∗}s of Ṫ M̃_{(ℓ)} and a section f^{∗}σ of V_{2}(T M̃ ⊕ ϵ)_{(ℓ)}. Then, because f is a double cover, deg(f^{∗}s) = 2 deg(s), and by the naturality of ϕ_{r} and (2.2) we have that Φ_{r}^{ℓ}(f^{∗}σ)(f^{∗}s) = f^{∗}(Φ_{r}^{ℓ}(σ)(s)). On the other hand, since M̃ is orientable, by the previous paragraph we know that deg(Φ_{r}^{ℓ}(f^{∗}σ)(s)) = r deg(f^{∗}s) − (r − 1) deg(f^{∗}σ_{0}). As a consequence:

2 deg(Φ_{r}^{ℓ}(σ)(s)) = deg(f^{∗}(Φ_{r}^{ℓ}(σ)(s)))
= deg(Φ_{r}^{ℓ}(f^{∗}σ)(f^{∗}(s))
= r(deg(f^{∗}s) − deg(f^{∗}σ_{0})) + deg(f^{∗}σ_{0})
= r(2 deg(s) − 2 deg(σ_{0})) + 2 deg(σ_{0}).  □

We now face the following lifting problem:

\xymatrix{ & V_{2}(T M ⊕ ϵ)_{(ℓ)} \ar[d]^{φ_{(ℓ)}} \\ M \ar[r]^{σ_{0}} \ar@{-->}[ur]^{σ} & S(T M ⊕ ϵ)_{(ℓ)},}

Proposition 2.6 Let M be closed and of dimension n ⩾ 2. When n is odd every diagram has a lift, and when n is even the diagram has a lift precisely for sections σ_{0} of degree χ/2 (which exist whenever χ is even or 2 ∉ ℓ).

Proof. The above problem is equivalent to find a section of the pullback η_{(ℓ)} of φ_{(ℓ)} along σ_{0}, which is an S_{(ℓ)}^{n−1}-bundle over an n-dimensional manifold. If n is odd, η_{(ℓ)} has always a section, hence in that case every section σ_{0} admits a lift. If n is even, the complete obstruction (if M is orientable) is the Euler class e(η_{(ℓ)}) of η_{(ℓ)}. We proceed to compute it: Assume first that M is orientable and ℓ = Spec ℤ. The bundle η is the unit sphere bundle of σ_{0}^{∗}T^{v}(T M ⊕ ϵ), whose Euler number can be computed by taking the self-intersection of its zero section in the fibrewise one point compactification of σ_{0}^{∗}T^{v}(T M ⊕ ϵ), which is precisely S(T M ⊕ ϵ). As the zero section of σ_{0}^{∗}T^{v}(T M ⊕ ϵ) is σ_{0}, we have that (we denote by x^{∨} the Poincaré dual of x)

(2.3) e(η)^{∨} = σ_{0}[M] ∩ σ_{0}[M]

(2.4) = (deg(σ_{0}) − χ, 1) (\begin{matrix}0 & 1 \\ 1 & χ \end{matrix}) (\begin{matrix}deg(σ_{0}) − χ \\ 1 \end{matrix}) = 2 deg(σ_{0}) − χ.

Hence a section admits a lift if and only if deg(σ_{0}) = χ/2. Let us assume now that M is orientable and ℓ is a proper subset of Spec ℤ. In this case, the above computation is no longer valid, as it relies on a geometric interpretation of the Euler class. We will first compute the Euler class e of φ_{(ℓ)}:

e(η_{(ℓ)}) = σ_{0}^{∗}(e)^{∨} = e ⌢ σ_{0}[M] = e^{∨} ∩ σ_{0}[M],

and therefore, if e^{∨} = (a, b) in the basis described before, it holds that

e(η_{(ℓ)})^{∨} = σ_{0}^{∗}(e)^{∨} = (a, b) (\begin{matrix}0 & 1 \\ 1 & χ \end{matrix}) (\begin{matrix}deg(σ_{0}) − χ \\ 1 \end{matrix}) = a + b deg(σ_{0}).

This, together with (2.3) (which holds for integral values), implies that e^{∨} = (−χ, 2), and therefore that σ_{0}^{∗}(e)^{∨} = 2 deg(σ_{0}) − χ. Hence, after localising we obtain that only sections of degree χ/2 admit a lift.

Finally, let M be non-orientable and let f: M̃ → M be the orientation cover of M. Then deg_{f^{∗}σ_{0}}(f^{∗}σ_{0}) = 2 deg_{σ_{0}}(σ_{0}) and the Euler characteristic of M̃ is 2χ, so deg_{σ_{0}}(σ_{0}) = 0 if and only if (2χ)/2 = deg(f^{∗}σ_{0}) = 2 deg(σ_{0}). Hence only sections of degree χ/2 have lifts. □

Combining Lemma 2.5 and Proposition 2.6, we have the following (see immediately above Lemma 2.1 for the notation [f]).

Corollary 2.7 Let ℓ be a collection of primes. Suppose that dim(M) is odd and we are given any r, d ∈ ℤ. Then there exists an endomorphism f ∈ End_{c}^{r}(Ṫ M_{(ℓ)}) with [f](k) = r(k − d) + d. Suppose that dim(M) is even and we are given any r ∈ ℤ. Assume also that χ/2 ∈ ℤ_{(ℓ)}, i.e., either χ is even or 2 ∉ ℓ. Then there exists an endomorphism f ∈ End_{c}^{r}(Ṫ M_{(ℓ)}) with [f](k) = r(k − χ/2) + χ/2.

2.4. Proof of Theorem A. As promised in the introduction, in the next three propositions we will provide the middle map in the zigzag (1.2), from which the assertions in the theorem will follow by virtue of Lemma 2.1.

Proposition 2.8 If dim M is odd and ℓ is any set of odd primes, then there are homotopy equivalences

Γ_{c}(Ṫ M)_{k} ⟶^{≃} Γ_{c}(Ṫ M)_{k+2}
Γ_{c}(Ṫ M_{(ℓ)})_{k} ⟶^{≃} Γ_{c}(Ṫ M_{(ℓ)})_{k+1}.

Proof. By Corollary 2.7 and Theorem 3.3 of [Dold, 1963] (cf. the discussion above Lemma 2.1), there exist homotopy equivalences

Γ_{c}(Ṫ M_{(ℓ)})_{k} ⟶ Γ_{c}(Ṫ M_{(ℓ)})_{r(k−d)+d}

for all integers r and d such that r ∉ ℓℤ (if ℓ = Spec ℤ), then the condition becomes that r = 1, −1). Observe first that if r is odd, then k and r(k − d) + d have the same parity. Hence if k and j have different parity then a homotopy equivalence Γ_{c}(Ṫ M_{(ℓ)})_{k} ⟶ Γ_{c}(Ṫ M_{(ℓ)})_{j} as above exists only if 2 ∉ ℓ. Taking r = −1 and d = k +1 (and ℓ = Spec ℤ) we obtain the first map. Taking r = 2 and d = k − 1 we obtain the second map. □

Proposition 2.9 Suppose that dim M is even, ℓ is a set of primes and k, j are integers such that (2k − χ)_{ℓ} = (2j − χ)_{ℓ}. If χ is odd, assume also that 2 ∉ ℓ. Then there is a zigzag of homotopy equivalences between Γ_{c}(Ṫ M_{(ℓ)})_{k} and Γ_{c}(Ṫ M_{(ℓ)})_{j}. If j = χ − k, there is a homotopy equivalence Γ_{c}(Ṫ M)_{k} → Γ_{c}(Ṫ M)_{j}.

Proof. By Corollary 2.7 and Theorem 3.3 of [Dold, 1963], there exists, for each integer r with trivial ℓ-adic valuation, a homotopy equivalence

Γ_{c}(Ṫ M_{(ℓ)})_{k} ⟶ Γ_{c}(Ṫ M_{(ℓ)})_{r(k−χ/2)+χ/2}.

Equivalently, there exists a homotopy equivalence Γ_{c}(Ṫ M_{(ℓ)})_{k} → Γ_{c}(Ṫ M_{(ℓ)})_{j} whenever (2j −χ) = r(2k−χ) for some r such that (r)_{ℓ} = 0. Now let k and j be the two given integers, let l = (2k − χ)_{ℓ} = (2j − χ)_{ℓ} and define m = ∏_{p∈ℓ} p^{l(p)}. Note that the integer \frac{1}{m}(2k − χ)(2j − χ) + χ is always even, so we have

\frac{1}{m}(2k − χ)(2j − χ) = (2h − χ)

for some integer h. Since \frac{1}{m}(2j − χ) and \frac{1}{m}(2k − χ) both have trivial ℓ-adic valuation, the previous discussion implies that there are homotopy equivalences Γ_{c}(Ṫ M_{(ℓ)})_{k} → Γ_{c}(Ṫ M_{(ℓ)})_{h} and Γ_{c}(Ṫ M_{(ℓ)})_{j} → Γ_{c}(Ṫ M_{(ℓ)})_{h}. For the last claim, observe that if M has even Euler characteristic χ, taking r = −1 and ℓ = Spec ℤ in Corollary 2.7, we obtain a homotopy equivalence Γ_{c}(Ṫ M) → Γ_{c}(Ṫ M) (without localising) that sends sections of degree k to sections of degree χ − k. In fact, such a homotopy equivalence exists regardless of whether χ is even or odd: it may be obtained by postcomposition with the antipodal map Ṫ M → Ṫ M. □

If dim M is odd, Theorem A can only be improved when 2 ∈ ℓ and k, j have different parity. To face this problem using a zigzag as in (1.2), we need to find a fibrewise homotopy equivalence f of Ṫ M_{(2)} whose action on components of the section space changes the parity. The following proposition deals with this case (cf. [Hansen, 1974; Hansen, 1981]). We note that its proof can be used to recover Theorem A when M is a sphere, as well as the integral homology isomorphisms of Theorem A when M is an arbitrary odd-dimensional manifold (see Remark 2.11).

Proposition 2.10 For n odd and 2 ∈ ℓ, the fibre bundle Ṫ S_{(ℓ)}^{n} admits fibrewise homotopy equivalences that change the parity of the sections if and only if n = 1, 3, 7. If dim M = 1, 3, 7, then Ṫ M admits a fibrewise homotopy equivalence that sends sections of degree k to sections of degree k + 1.

Proof. Spheres are stably parallelisable, and therefore Ṫ S^{n} is trivial, being the unit sphere bundle of T S^{n} ⊕ ϵ. After choosing a trivialisation, a fibrewise endomorphism f of Ṫ S_{(ℓ)}^{n} of degree r is the same as a map

f^{t} : S^{n} ⟶ Map_{r}(S_{(ℓ)}^{n}, S_{(ℓ)}^{n}).

Since n is odd, the Euler characteristic χ is 0, so we may trivialise Ṫ S^{n} so that the zero section corresponds to a trivial section of the product bundle S^{n} ×S_{(ℓ)}^{n}. Hence the matrix of H_{n}(f) with the basis considered in Lemma 2.3 is of the form

(\begin{matrix}r & b \\ 0 & 1 \end{matrix}) ,

where b ∈ ℤ_{(ℓ)} is the degree of the composition of f^{t} with the evaluation map. Such an f sends sections of degree k to sections of degree rk + b. For f to be a fibrewise homotopy equivalence, r must be odd (and moreover equal to ±1 when ℓ = Spec ℤ), and therefore b has to be odd as well, because [f](k) = rk+b and we want k and [f](k) to have different parity. Therefore, we need to solve the lifting problem

\xymatrix{ & Map_{r}(S_{(ℓ)}^{n}, S_{(ℓ)}^{n}) \ar[d] \\ S^{n} \ar[r]^{b} \ar@{-->}[ur] & S_{(ℓ)}^{n}}

where the vertical map is the evaluation map and the horizontal map is some map of odd degree b. The single (and therefore complete) obstruction to the existence of a lift is a class in H^{n}(S^{n}; π_{n−1}Ω_{r}^{n}S_{(ℓ)}^{n}) ≅ π_{n−1}Ω_{r}^{n}S_{(ℓ)}^{n}. This class is b times the image of the generator of π_{n}S_{(ℓ)}^{n} under the boundary homomorphism in the long exact sequence of homotopy groups. The boundary homomorphism is computed in [Whitehead, 1946, Theorem 3.2] with a correction in [Whitehead, 1953], who proved that under the identification π_{i}(Ω_{r}^{n}S_{(ℓ)}^{n}) ≅ π_{n+i}(S_{(ℓ)}^{n}), it corresponds to taking the Whitehead product with −rι, where ι is a generator of π_{n}(S^{n}). Therefore our obstruction is b[−rι, ι]. Because the Whitehead product is graded-commutative, [ι, ι] has order two, so −br[ι, ι] = [ι, ι]. The EHP sequence shows that the vanishing of this class is equivalent to the existence of elements of Hopf invariant one in π_{2n+1}(S_{(ℓ)}^{n+1}), which exist if and only if n = 0, 1, 3, 7 [Adams, 1960]. If M is an arbitrary manifold of dimension 1, 3 or 7, we first choose an open disc D in M. Then we take ℓ = ∅, r = 1, b = 1, and we consider the onepoint compactification Ḋ of D. By the previous part, there is a fibrewise endomorphism f of fibrewise degree 1 of Ṫ Ḋ. Without loss of generality, we may assume that its value on the basepoint is the identity. Then we can extend this fibrewise endomorphism to the whole of M by defining x ↦ (Id: Ṫ_{x}M → Ṫ_{x}M) if x ∉ D. The extension sends sections of degree k to sections of degree rk + b = k + 1. By Theorem 3.3 of [Dold, 1963], it is a fibrewise homotopy equivalence. □

Remark 2.11 The above proof recovers Theorem A in certain cases, as follows. Given some k ∈ ℤ, we take r = 2 and b = 1−k to obtain homotopy equivalences Γ_{c}(Ṫ S_{(ℓ)}^{n})_{k} ≃ Γ_{c}(Ṫ S_{(ℓ)}^{n})_{k+1} with ℓ the set of all odd primes. This recovers the isomorphisms (1.5) when M = S^{n}. For the isomorphisms (1.4), we take r = 1 and b = 2 (and ℓ the set of all primes, i.e., we do not localise) and extend the resulting fibrewise homotopy equivalence, defined over a disc in M, to the whole of M by the identity. This gives homotopy equivalences Γ_{c}(Ṫ M)_{k} ≃ Γ_{c}(Ṫ M)_{k+2} for any M.

Proposition 2.10 also shows that, when M is an odd-dimensional sphere, Theorem A cannot be improved by finding a fibrewise homotopy equivalence of Ṫ S_{(ℓ)}^{n}. The following proposition (cf. [Hansen, 1974, Theorem 3.1] and [Hansen, 1981]) generalises the computation of H_{1}(C_{k}(S^{2}); ℤ) (which follows from the presentation of π_{1}(C_{k}(S^{2})) given by [Fadell and Van Buskirk, 1962]) and shows that Theorem A is sharp when M is an even-dimensional sphere.

Proposition 2.12 If n is even and k belongs to the stable range with respect to homological degree n − 1, then

H_{n−1}(C_{k}(S^{n}); ℤ) ≅ τH_{n−1}(Ω_{0}^{n}S^{n}) ⊕ ℤ/(2k − 2),

where τG is the torsion of G. If M is a closed manifold of even dimension, then any fibrewise endomorphism of Ṫ M_{(ℓ)} of degree r ≠ 0 sends sections of degree k to sections of degree r(k − χ/2) + χ/2.

Proof. The target of the scanning map in this case is Γ(Ṫ S^{n}), and since S^{n} is stably parallelisable, Ṫ S^{n} can be trivialised. The trivialisation gives a homotopy equivalence Γ(Ṫ S^{n}) → Map(S^{n}, S^{n}) that sends sections of degree k to maps of degree k − χ/2, where χ is the Euler characteristic of S^{n} (this corresponds to a change of basis in Lemma 2.3, and is stated explicitly in [Bendersky and Miller, 2014, Proposition 3.6]). We let now r = k − χ/2. The space of maps fits into the evaluation fibration

(2.5)  Ω_{r}^{n}S^{n} ⟶ Map_{r}(S^{n}, S^{n}) ⟶ S^{n}

for which we may consider the corresponding Wang sequence

⋯ ⟶ H_{0}(Ω_{r}^{n}S^{n}) ⟶^{δ_{r}} H_{n−1}(Ω_{r}^{n}S^{n}) ⟶ H_{n−1}(Map_{r}(S^{n}, S^{n})) ⟶ 0.

The map named δ_{r} is the transgression

H_{n}(S^{n}; H_{0}(Ω_{r}^{n}S^{n})) ⟶ H_{0}(S^{n}; H_{n−1}(Ω_{r}^{n}S^{n}))

in the Serre spectral sequence of the evaluation fibration, and therefore under the identification H_{n}(S^{n}; H_{0}(Ω_{r}^{n}S^{n})) = H_{n}(S^{n}) it fits into the commutative diagram

\xymatrix{π_{n}(S^{n}) \ar[d] \ar[r]^{∂_{r}} & π_{n−1}(Ω_{r}^{n}S^{n}) \ar[d] \\ H_{n}(S^{n}) \ar[r]^{δ_{r}} & H_{n−1}(Ω_{r}^{n}S^{n}),}

where ∂_{r} is the boundary homomorphism in the long exact sequence of homotopy groups. As recalled in the previous proof, ∂_{r} was identified by Whitehead as the adjoint of the operation of taking Whitehead product with −rι, where ι is the generator of S^{n}. Additionally, the left vertical arrow is an isomorphism and the rightmost vertical arrow sends the class [ι, ι] to the Browder square of the generator of H_{0}(Ω_{r}^{n}S^{n}) (see Remark 1.2 in the third chapter of [Cohen et al., 1976]) (here we are using a canonical identification of Ω_{r}^{n}S^{n} and Ω_{1}^{n}S^{n} to define the Browder square and the Whitehead product). We claim, when n is even, that this Browder square has infinite order and is divisible by two (but not by four). To see this, consider the scanning map C(ℝ^{n}) → Ω^{n}S^{n}. Both spaces have an action of the little n-discs operad, and the scanning map is equivariant with respect to this action, hence it takes Browder squares to Browder squares. The adjoint α of the class ι ∈ π_{n}(S^{n}) is a generator of π_{0}(Ω_{1}^{n}S^{n}), so the Browder square λ(β, β) of the Hurewicz image β of α lives naturally (that is, before using the identification between the different components of Ω^{n}S^{n}) in H_{n−1}(Ω_{2}^{n}S^{n}). We will now describe this class. The generator γ of H_{0}(C_{1}(ℝ^{n})) is mapped under the scanning map to β, hence λ(γ, γ) is mapped to λ(β, β). The class λ(γ, γ) ∈ H_{n−1}(C_{2}(ℝ^{n})) corresponds to moving one of the points around the other point in all possible directions, parametrised by S^{n−1}. The inclusion of the space ℝP^{n−1} of antipodal points in S^{n−1} into C_{2}(ℝ^{n}) is a homotopy equivalence, and our class λ(γ, γ) is the image of the fundamental class of S^{n−1} under the double covering map S^{n−1} → ℝP^{n−1}, hence it is twice a generator of the group H_{n−1}(C_{2}(ℝ^{n})) ≅ H_{n−1}(ℝP^{n−1}) ≅ ℤ (cf. the class τ on page 780). Now, since the scanning map is split-injective on homology (see [McDuff, 1975, p. 103] and the proof of Corollary 3.2 in this article), it follows that λ(β, β) also has infinite order and is divisible exactly by two. Observe that, when n is odd, the above argument shows that λ(β, β) is zero, since ℝP^{n−1} is non-orientable for n − 1 even. By results of Serre [Serre, 1951] on the homotopy groups of spheres, and the rational Hurewicz theorem, H_{n−1}(Ω_{r}^{n}S^{n}) has rank 1, so

H_{n−1}(Map_{r}(S^{n}, S^{n})) ≅ H_{n−1}(Ω_{r}^{n}S^{n})/(−rλ(β, β)) ≅ τH_{n−1}(Ω_{r}^{n}S^{n}) ⊕ ℤ/2r.

The first statement now follows from McDuff’s theorem. Let f: Ṫ M_{(ℓ)} → Ṫ M_{(ℓ)} be any fibrewise endomorphism of Ṫ M_{(ℓ)}, where we view the bundle Ṫ M_{(ℓ)} → M as a fibration. We can consider the fibrewise rationalisation f_{(0)} : Ṫ M_{(0)} → Ṫ M_{(0)}, and observe that [f](k) = [f_{(0)}](k) for k ∈ ℤ_{(ℓ)} (using the canonical inclusion of ℤ_{(ℓ)} in ℚ). Therefore the function [f], describing the effect of f on degrees of sections of Ṫ M_{(ℓ)}, is determined by the function [f_{(0)}]. If dim M is even, there is a unique fibrewise endomorphism of Ṫ M_{(0)} of fibrewise degree r up to homotopy. This is because such fibrewise endomorphisms are sections of a bundle over M with fibre Map_{r}(S_{(0)}^{n}, S_{(0)}^{n}), which, using the evaluation fibration (2.5), is (2n − 2)-connected since [ι, rι] ≠ 0. Therefore, if f and g are fibrewise endomorphisms of Ṫ M_{(ℓ)} with the same (non-zero) fibrewise degree, we have that [f](k) = [f_{(0)}](k) = [g_{(0)}](k) = [g](k) for all k ∈ ℤ_{(ℓ)}, so they act in the same way on the path-components of the section space. Let r = p/q be a non-zero rational number, with p, q ∈ ℤ ∖ {0}, and let f_{r} be any fibrewise endomorphism of Ṫ M_{(0)} of degree r. Since p and q are integers, Corollary 2.7 implies that there are fibrewise endomorphisms ϕ_{p} and ϕ_{q} of degrees p and q respectively. Let f_{p} = ϕ_{q}f_{r}, which is a fibrewise endomorphism of degree p. By the previous paragraph, there is a unique fibrewise endomorphism of each degree, so it follows that f_{p} is fibrewise homotopic to ϕ_{p}. We therefore have the equation:

[ϕ_{q}][f_{r}](k) = [ϕ_{p}](k).

The last formula of Corollary 2.7 determines the functions [ϕ_{p}] and [ϕ_{q}], from which we deduce that

q([f_{r}](k) − χ/2) + χ/2 = p(k − χ/2) + χ/2 and the result follows after solving the equation. □

3. The extrinsic replication map

3.1. Scanning maps. Let M be a connected manifold, for which we choose a Riemannian metric with injectivity radius bounded below by δ > 0. Let T^{1}M denote the open unit disc bundle of the tangent bundle of M, and let Ṫ^{1}M and Ṫ M denote the fibrewise one point compactifications of T^{1}M and T M. Let δ > 0 be smaller than the injectivity radius of M. Define the linear scanning map

𝒮 : C_{k}^{δ}(M) ⟶ Γ_{c}(Ṫ^{1}M)_{k}

to the space of degree k compactly supported sections of Ṫ^{1}M as

𝒮 (q, ϵ)(x) = \begin{cases}∞ & if x ∉ B_{ϵ}(q) ∀q ∈ q, \\ \frac{exp_{x}^{−1}(q)}{ϵ} & if x ∈ B_{ϵ}(q), q ∈ q. \end{cases}

The degree of a section s is the fibrewise intersection, counted with multiplicity, of s and the zero section (see also §2.2). Let D be the unit n-dimensional open disc, let Ḋ be its one point compactification and define ψ^{δ}(D) to be the quotient of ⋃_{k} C_{k}^{δ}(ℝ^{n}), where two configurations (q, ϵ) and (q′, ϵ′) are identified if q ∩ D = q′ ∩ D and either ϵ = ϵ′ or q ∩ D = ∅. We write ψ^{δ}(T^{1}M) for the result of applying this construction fibrewise to T^{1}M. Let γ be a number smaller than the injectivity radius of M. The radius γ non-linear scanning map

𝔰: C_{k}^{δ}(M) → Γ_{c}(ψ^{δ}(T^{1}M))

sends a configuration q to \frac{1}{γ} exp_{x}^{−1}(q) — which may consist of more than one point. There is an inclusion i: Ḋ ↪ ψ^{δ}(D) given by i(q) = (q, δ/2) as the subspace of configurations with at most one point. This inclusion has a homotopy inverse

h(q, ϵ) = \frac{q}{q_{second}}

where q_{first} is the norm of a closest point in q to the origin, and q_{second} is defined to be 1 if |q| = 1 and (q′)_{first} otherwise, where q′ is the result of removing a single closest point of q to the origin. The composite hi is the identity and H_{t}(q, ϵ) = (\frac{q}{(1−t)+tq_{second}} , tδ/2 + (1 − t)ϵ) gives a homotopy between the identity and ih. Each of i, h and H_{t} is O(n)-equivariant, so they can be defined on the vector bundle T M, obtaining homotopy equivalences

i: Ṫ^{1}M ⟷ ψ^{δ}(T^{1}M): h

which induce by composition homotopy equivalences

i: Γ_{c}(Ṫ^{1}M) ⟷ Γ_{c}(ψ^{δ}(T^{1}M)): h

that commute with the linear and non-linear scanning maps:

(3.1)  _{C_{k}^{δ}(M) Γ_{c}(ψ^{δ}T^{1}M) Γ_{c}(Ṫ^{1}M). 𝔰^{γ} 𝒮 h i≃}

3.2. Homological stability.

Theorem B. Let M be a connected, smooth manifold and let v be a nonvanishing section of T M. Then there exists a map ϕ_{r} ∈ End_{c}^{r}(Ṫ^{1}M) that makes the following diagram commute up to homotopy:

\begin{matrix}C_{k}^{δ}(M) & 𝒮 & [figure] & ρ_{r}[v] & [figure] & Γ_{c}(Ṫ^{1}M → M)_{k} & ϕ_{r} & [figure] & C_{rk}^{δ}(M) & 𝒮 & [figure] & Γ_{c}(Ṫ^{1}M → M)_{rk}. \\ (3.2) \end{matrix}

Hence the r-replication map induces an isomorphism on ℤ_{(ℓ)}-homology in the stable range with ℤ_{(ℓ)} coefficients as long as r is not divisible by any prime in ℓ.

Remark 3.1 One can prove that the map ϕ_{r} constructed below is homotopic to Φ_{r}^{ℓ}(ι, v) with ℓ = Spec ℤ.

Proof. The proof has three steps. First, since C_{k}^{δ}(M) is independent of δ up to homotopy, we let 2δ be smaller than the injectivity radius of M. We claim that the following diagram commutes:

\xymatrix{C_{k}^{δ}(M) \ar[r]^{𝔰^{2δ}} \ar[d]^{ρ_{r}[v]} & Γ_{c}(ψ^{δ}(T^{1}M)) \ar[d]^{ς_{r}} \\ C_{rk}^{δ}(M) \ar[r]^{𝔰^{δ}} & Γ_{c}(ψ^{δ}(T^{1}M))}

where ς_{r} is given by postcomposition with the bundle map ρ_{r}[exp_{2δ}^{∗}(v)]: ψ^{δ}(T^{1}M) → ψ^{δ}(T^{1}M) followed by the expansion 𝟚: ψ^{δ}(T^{1}M) → ψ^{δ}(T^{1}M) that sends each point q in the configuration to 2q. Observe that the bundle map ρ_{r}[exp_{2δ}^{∗}(v)] is not continuous but it becomes continuous after composing with 𝟚. In order to understand this square, we check what happens with the adjoint of the scanning map M × C_{k}(M) → ψ^{δ}(T^{1}M) over each point x ∈ M:

\xymatrix{\{x\} × C_{k}^{δ}(M) \ar[r]^{𝔰_{x}^{2δ}} \ar[d]^{ρ_{r}[v]} & ψ^{δ}(T_{x}^{1}M) \ar[d]^{ς_{r}} \\ \{x\} × C_{rk}^{δ}(M) \ar[r]^{𝔰_{x}^{δ}} & ψ(T_{x}^{1}M).}

The square commutes on the nose unless there exists some q ∈ q such that

ς_{r}(q) ∩ B_{δ}(x) ≠ ∅, and q ∉ B_{2δ}(x).

But this is not possible, as d(ς_{r}(q), x) ⩾ d(q, x) − max_{q′∈ς_{r}(q)} d(q, q′) ⩾ 2δ − ϵ ⩾ δ. Second, observe that since the exponential map is homotopic to the projection π: T M → M, the maps ς_{r} = 𝟚ρ_{r}[exp_{2δ}^{∗}(v)] and σ_{r} = 𝟚ρ_{r}[π^{∗}v] are homotopic. Third, consider now the diagram

\xymatrix{Γ_{c}(ψ^{δ}(T^{1}M)) \ar[d]^{σ_{r}} & Γ_{c}(Ṫ^{1}M) \ar[d] \ar[l]_{i} \\ Γ_{c}(ψ^{δ}(T^{1}M)) \ar[r]^{h} & Γ_{c}(Ṫ^{1}M)}

whose maps are induced by the fibrewise maps which on each fibre are

\xymatrix{ψ^{δ}(T_{x}^{1}M) \ar[d]^{σ_{r}} & Ṫ_{x}^{1}M \ar[d] \ar[l]_{i} \\ ψ^{δ}(T_{x}^{1}M) \ar[r]^{h} & Ṫ_{x}^{1}M}

Let us denote by v the value of the vector field v at the point x. Then σ_{r}(q, 1) = q ∪ q + v ∪ … ∪ q + (r − 1)v and

hσ_{r}i(q) = \begin{cases}2\frac{q+jv}{∥q+(j−1)v∥} & if q + jv is the closest point and ⟨v, q + jv⟩ > 0 \\ 2\frac{q+jv}{∥q+(j+1)v∥} & if q + jv is the closest point and ⟨v, q + jv⟩ < 0. \end{cases}

The inverse image of a point (for instance the origin) consists of r points ({−jv}_{j=0}^{r−1}), all of them oriented according to the sign of r. Hence hσ_{r}i induces a map of degree r on fibres. □

Corollary 3.2 If M is a connected open manifold of dimension at least 2, then the homomorphism induced on ℤ[\frac{1}{r}]-homology by the r-replication map is split-injective.

Proof. The scanning map is split-injective on homology in all degrees, as may be deduced from [McDuff, 1975]. To see this, recall the following facts from the referenced paper: (a) the stabilisation maps C_{k}(M) → C_{k+1}(M) are splitinjective on homology in all degrees (see p. 103), (b) the analogous stabilisation maps for section spaces Γ_{c}(Ṫ M)_{k} → Γ_{c}(Ṫ M)_{k+1} are homotopy equivalences,

(c) the scanning map C_{k}(M) → Γ_{c}(Ṫ M)_{k} is a homology equivalence in the colimit as k → ∞ and (d) the homology in the colimit is finitely-generated. It then follows from the fact that stabilisation and scanning commute that the homomorphism induced on homology by the scanning map is the composition of a finite sequence of split-injections, and therefore itself a split-injection.

Thus, in the commutative square (3.2), the composite ϕ_{r} ∘ 𝒮 is split-injective on ℤ[\frac{1}{r}]-homology, hence 𝒮 ∘ ρ_{r}[v] is split-injective on ℤ[\frac{1}{r}]-homology, and so ρ_{r}[v] is also split-injective on ℤ[\frac{1}{r}]-homology. □

4. The intrinsic replication map

4.1. Stabilisation, replication and scanning maps with labels. Let θ: E → M be a fibre bundle, and define the following spaces (the first two were also defined in the introduction; the third space is defined whenever γ is smaller than the injectivity radius of M, in particular when γ < δ):

C_{k}(M; θ) = {(q, f) | q ∈ C_{n}(M), f ∈ Γ(θ|_{q})}
C_{k}^{δ}(M; θ) = {(q, ϵ, f) | (q, ϵ) ∈ C_{k}^{δ}(M), f ∈ Γ(θ|_{B_{q}(ϵ)})}
C_{k}^{δ,γ}(M; θ) = {(q, ϵ, {f_{q}}_{q∈q}) | (q, ϵ) ∈ C_{k}^{δ}(M), f_{q} ∈ Γ(θ|_{B_{q}(γ)})}.

Lemma 4.1 The forgetful maps

C_{k}^{δ,γ}(M; θ) ⟶ C_{k}^{δ}(M; θ) ⟶ C_{k}(M; θ)

that restrict the section first from balls of radius γ to balls of radius ϵ, and then to the centres of the balls, are weak homotopy equivalences.

Proof. A point in C_{k}^{δ}(M; θ) consists of a configuration q with prescribed pairwise separation, together with a choice of label on a small contractible neighbourhood of each configuration point. On the other hand, the pullback Ċ_{k}^{δ}(M; θ) of C_{k}^{δ}(M) along the map C_{k}(M; θ) → C_{k}(M) which forgets the labels consists of a configuration with prescribed pairwise separation, together with a choice of label just over each configuration point. Since C_{k}^{δ}(M) → C_{k}(M) is a fibre bundle with contractible fibres, so is its pullback Ċ_{k}^{δ}(M; θ) → C_{k}(M; θ), which is therefore a weak equivalence. There is also a forgetful map C_{k}^{δ}(M; θ) → Ċ_{k}^{δ}(M; θ) which just remembers the label at the centre of each ball. This is also a fibre bundle with contractible fibres, so a weak equivalence. Hence the composition C_{k}^{δ}(M; θ) → C_{k}(M; θ) which completely forgets the labels (the second map in Lemma 4.1) is a weak equivalence. The first map in Lemma 4.1 is a fibre bundle with contractible fibres, so it is also a weak equivalence. □

Definition 4.2 If the manifold M is open, a choice of an embedding of a ray, together with a point y in the ray and a label f_{y} ∈ θ^{−1}(y) of this point defines a stabilisation map with labels

s^{θ}: C_{k}(M; θ) ⟶ C_{k+1}(M; θ)

by pushing the configuration outside the ray and adding the labelled point (y, f_{y}).

Definition 4.3 If the manifold M is open or has trivial Euler characteristic, a choice of a non-vanishing vector field defines a replication map with labels

ρ_{r}^{θ}: C_{k}^{δ}(M) ⟶ C_{rk}^{δ}(M; θ)

ρ_{r}^{θ,γ}: C_{k}^{δ,γ}(M; θ) ⟶ C_{rk}^{δ,γ/r}(M; θ)

by sending a configuration ((q, ϵ), f) to the configuration ρ_{r}(q, ϵ) together with the restriction of the section f to the balls of radius \frac{ϵ}{2r} (and radius \frac{γ}{r}) centered at the points in the new configuration.

The pullback θ^{∗}T M → E is also fibred over M, and the fibres are vector bundles. We denote by Ṫ^{θ}M the fibrewise Thom construction of θ^{∗}T M viewed as a bundle over M. The inclusion of the points at infinity define a cofibre sequence over M

(4.1)  E ⟶ θ^{∗}Ṫ M ⟶ Ṫ^{θ}M.

The pullback map θ^{∗}Ṫ M → Ṫ M factors through the bundle maps

(4.2)  θ^{∗}Ṫ M ⟶ Ṫ^{θ}M ⟶^{ξ} Ṫ M.

We define the degree of a section s as the degree of ξ(s). If the fibres of θ are path connected, then the n-skeleton of the fibres of Ṫ^{θ}M is homotopic to S^{n}, therefore the forgetful map

Γ_{c}(Ṫ^{θ}M) ⟶ Γ_{c}(Ṫ M)

induces a bijection on connected components. We write Ṫ^{1,θ}M for the analogous construction with T^{1}M.

Definition 4.4 Define the linear scanning map

𝒮^{θ}: C_{k}^{δ}(M; θ) ⟶ Γ(Ṫ^{1,θ}M)

by sending a configuration (q, ϵ, f) to the section whose value at a point x ∈ M is 𝒮 (q, ϵ)(x) together with the label f|_{x} if 𝒮 (q, ϵ)(x) ≠ ∞.

Let D be the unit n-dimensional open disc, let Ḋ be its one-point compactification and let F be a space. Define ψ^{δ}(D; F) to be the quotient of the space ⋃_{k} C_{k}^{δ}(ℝ^{n}; θ: F × ℝ^{n} → ℝ^{n}), where two labeled configurations (q, ϵ, f) and (q′, ϵ′, f′) are identified if q∩D = q′∩D, f|_{B_{ϵ}(q)} = f′|_{B_{ϵ}(q)} for all q ∈ q∩D, and either q ∩ D = ∅ or ϵ = ϵ′. Let ψ^{δ}(T^{1}M; θ) be the result of applying this construction fibrewise to the unit ball of the tangent bundle of M and the fibre bundle θ, so that

ψ^{δ}(T^{1}M; θ) = ⋃_{x∈M} ψ^{δ}(T_{x}^{1}M; θ^{−1}(x)).

Definition 4.5 The non-linear scanning map with labels in θ

𝔰^{θ,γ}: C_{k}^{δ,γ}(M; θ) ⟶ Γ(ψ^{δ}(T^{1}M; θ))

sends a configuration (q, ϵ, {f_{q}}_{q∈q}) to the section that assigns to the point x ∈ M the triple (q′, ϵ′, {f′}_{q′∈q′}), where (q′, ϵ′) = 𝔰(q, ϵ) ∈ ψ^{δ}(T_{x}^{1}M), and f′_{q′} is constant with value f_{q}(x) if q′ = \frac{1}{γ} exp^{−1}(q).

There are fibrewise maps

i: Ṫ^{1,θ}M ↪ ψ^{δ}(T^{1}M; θ)
h: ψ^{δ}(T^{1}M; θ) ⟶ Ṫ^{1,θ}M

defined by i(∞) = ∞, h(∞) = ∞ and

i(q, y) = (q, δ/2, f)  with f constant with value y

h(q, ϵ, \{f_{q}\}_{q∈q}) = \begin{cases}(\frac{q}{q_{second}} , f (q)) & if q_{first} < q_{second} \\ & and q ∈ q with ∥q∥ = q_{first}, \\ ∞ & if q_{first} = q_{second} \end{cases}

which are mutually fibrewise homotopy inverses by the same argument as on page 771 (where the definition of q_{second} is also given). In Appendix B we show that the stabilisation map with labels induces an isomorphism in a range ∗ ⩽ μ[M; θ](k), and we give lower bounds for it in Proposition B.3 and Remark B.4 – these are either the same or one degree less than the lower bounds given on page 754 for unlabelled configuration spaces. In Appendix C we give a proof of McDuff’s theorem with labels, as follows (where ν[M; θ] is the range in which the scanning map for M is an isomorphism on homology):

Theorem 4.6 Let θ: E → M be a fibre bundle with path-connected fibres. If M is non-compact then we have

ν[M; θ](k) = min_{j⩾k}{μ[M; θ](j)}.

The inequality ν[M; θ] ⩾ ν[M ∖ {∗}; θ|_{M∖{∗}}] holds for all M, so the function ν[M; θ] diverges for all M.

4.2. Homological stability. The fibrewise homotopy equivalences of Lemma 2.5 lift to fibrewise homotopy equivalences of θ^{∗}Ṫ^{1}M_{(p)}, which in turn descend to fibrewise homotopy equivalences Ṫ^{1,θ}M_{(p)} if and only if they fix the section at infinity (c.f. cofibration (4.1)). This implies that σ_{0} = ι in that lemma, and therefore each of these fibrewise homotopy equivalences sends sections of degree k to sections of degree rk. Hence we only recover part of Theorem A and the whole of Theorem B:

Theorem A′ If M is a closed, connected manifold with trivial Euler characteristic, then H_{∗}(C_{k}(M; θ)) ≅ H_{∗}(C_{j}(M; θ)) in the stable range with labels if k and j have the same p-adic valuation.

Theorem B′ If v is a non-vanishing vector field on a connected manifold M and p ∤ r, then the r-replication map ρ_{r}^{θ} with labels induces isomorphisms in ℤ_{(p)}-homology in the stable range with labels.

Proof. The relevant diagram is the following:

\xymatrix{C_{k}^{δ,γ} \ar[r]^{𝔰^{θ,γ}} \ar[d]^{ρ_{r}} & Γ_{c}(ψ^{δ}(T^{1}M; θ)) \ar[d] & Γ_{c}(Ṫ^{1,θ}M) \ar[d] \ar[l]_{i} \\ C_{rk}^{δ,γ/r} \ar[r]^{𝔰^{θ,γ/r}} & Γ_{c}(ψ^{δ}(T^{1}M; θ)) \ar[r]^{h} & Γ_{c}(Ṫ^{1,θ}M)}

The argument in the proof of Theorem B generalizes step by step to give that the left hand-side square commutes up to homotopy and that the rightmost vertical arrow is given by postcomposition with a bundle map which on the fibre over x ∈ M induces a map

\xymatrix{F \ar[r] \ar[d]^{Id} & F × S^{n} \ar[d]^{Id×f_{r}} \ar[d] \ar[r] & Σ^{n}F_{+} \ar[d] \\ F \ar[r] & F × S^{n} \ar[r] & Σ^{n}F_{+}}

where F = θ^{−1}(x), f_{r} is a degree r map, hence the rightmost vertical map is a ℤ[\frac{1}{r}]-homotopy equivalence. degree r between sphere bundles. □

Let θ be the projection S(T M) → M. Recall the definition of the intrinsic replication map � _{r}: C_{k}^{δ}(M; θ) ⟶ C_{rk}^{δ}(M; θ) from page 757.

Theorem C If M is a connected smooth manifold and ℓ is a set of primes not dividing an integer r, then the map � _{r}: C_{k}^{δ}(M; θ) → C_{rk}^{δ}(M; θ) induces isomorphisms on homology with ℤ_{(ℓ)}-coefficients in the stable range with labels.

Proof. Define σ_{r}: ψ(T M; θ) → ψ(T M; θ) to be the fibrewise version of � _{r} composed with 𝟚 as in the proof of Theorem B. The first square in the following diagram

\begin{matrix}C_{k}^{δ,γ}(M; θ) & 𝔰^{θ,γ} & [figure] & _{� _{r}} & [figure] & Γ_{c}(ψ^{δ}(T^{1}M; θ)) & σ_{r} & [figure] & Γ_{c}(Ṫ^{1,θ}M)_{k} & i & [figure] & hσ_{r}i & [figure] & C_{k}^{δ,γ/r}(M; θ) & 𝔰^{θ,γ/r} & [figure] & Γ_{c}(ψ^{δ}(T^{1}M; θ)) & h & [figure] & Γ_{c}(Ṫ^{1,θ}M)_{rk}. \\ (4.3) \end{matrix}

commutes, by the same argument as the first step the proof of Theorem B. Therefore we obtain a fibrewise map hσ_{r}i on the right hand side. The map hσ_{r}i is obtained by postcomposition with a fibrewise map g: Ṫ^{1,θ}M → Ṫ^{1,θ}M. The map g_{x} on the fibre over the point x restricts to the identity on the points at infinity, therefore it extends to the following diagram of cofibre sequences

\xymatrix{S(T_{x}M) × \{∞\} \ar[d]^{Id} \ar[r] & S(T_{x}M) × Ṫ_{x}^{1}M \ar[r] \ar[d]^{f} & Ṫ_{x}^{1,θ}M \ar[d]^{g_{x}} \\ S(T_{x}M) × \{∞\} \ar[r] & S(T_{x}M) × Ṫ_{x}^{1}M \ar[r] & Ṫ_{x}^{1,θ}M}

where the leftmost horizontal maps are the inclusion of the points at infinity. After localizing the diagram at ℓ, the map f_{(ℓ)} is a map of sphere bundles that induces a map of degree r on fibres (as in the proof of Theorem B). Since r is a unit in ℤ_{(ℓ)}, it follows that f_{(ℓ)} is a homotopy equivalence, and therefore that (g_{x})_{(ℓ)} is too. Since g_{(ℓ)} induces a homotopy equivalence on fibres, it follows (using Theorem 3.3 of [Dold, 1963]) that g_{(ℓ)} is a fibrewise homotopy equivalence, and therefore that (hσ_{r}i)_{(ℓ)} is a homotopy equivalence. □

If the bundle θ: E → M factors through S(T M) and has path-connected fibres (for instance the oriented frame bundle of T M, if M is orientable), then Theorem C generalises to:

Theorem C′ If ℓ is a set of primes, none of them dividing the integer r, then the intrinsic replication map with labels

� _{r}: C_{k}(M; θ) ⟶ C_{rk}(M; θ)

induces isomorphisms on homology with ℤ_{(ℓ)} coefficients in the stable range with labels.

5. Homological stability via vector fields with exactly one zero

We now use some different techniques to extend our results a bit further for homology with field coefficients. Section spaces are not involved in this part; instead we apply Theorem B (homological stability with respect to the rreplication map) to M ∖ {∗} and classical homological stability for M ∖ {∗} to obtain Theorem D.

5.1. Vector fields. Let M be a closed connected manifold with Euler characteristic χ.

Definition 5.1 Given a vector field v ∈ Γ(T M) with an isolated zero z ∈ M, define the degree deg_{v}(z) of z as follows. Choose a coordinate chart U ≅ ℝ^{n}, with z ∈ U corresponding to 0 ∈ ℝ^{n}, such that v has no other zeros in U. The differential of the diffeomorphism U ≅ ℝ^{n} is a bundle isomorphism T M|_{U} = T U ≅ T ℝ^{n} = ℝ^{n} × ℝ^{n}. The restriction of v to U ∖ {z} therefore determines a map ℝ^{n} ∖ {0} → ℝ^{n} ∖ {0}. The degree of this map is by definition deg_{v}(z).

A simple observation is that M admits a vector field with at most one zero, which will therefore have index χ by the Poincaré-Hopf theorem. Moreover we can choose exactly what this zero looks like locally:

Lemma 5.2 Suppose we are given a vector field v on a closed ball B ⊆ M with exactly one zero which lies in its interior and has index χ. Then this extends to a vector field v̂ on M which is non-vanishing on M ∖ B.

Proof. First choose a vector field w on M which has only isolated (and therefore finitely many) zeros. Choose a larger closed ball B′ ⊃ B and a trivialisation of T M|_{B′}. Now homotope w if necessary so that all its zeros lie in int(B′) ∖ B. The restriction of w to ∂B′ is a map ∂B′ → ℝ^{n} ∖ {0}, whose degree is the sum of the degrees of all zeros of w, which is χ by the Poincaré-Hopf theorem. The restriction of the vector field v to ∂B is a map ∂B → ℝ^{n} ∖ {0} which also has degree χ by assumption. Since any two maps S^{n−1} → ℝ^{n} ∖ {0} of the same degree are homotopic, there is a map x: B′ ∖int(B) ≅ S^{n−1} ×[0, 1] → ℝ^{n} ∖{0} agreeing with w on ∂B′ and with v on ∂B. We can therefore define v̂ to be equal to v on B, x on B′ ∖ B and w on M ∖ B′. □

5.2. A cofibre sequence of configuration spaces. Choose a Riemannian metric on M and an isometric embedding D ↪ M of the closed unit disc D ⊆ ℝ^{n}. Following [Randal-Williams, 2013a, §6] we define U_{k}(M) to be the subspace of C_{k}(M) of configurations which have a unique closest point in D to its centre 0 ∈ D. There is an open cover of C_{k}(M) given by the subsets U_{k}(M) and C_{k}(M ∖ {0}), with intersection U_{k}(M ∖ {0}). By excision, the induced map of mapping cones

(5.1)  (U_{k}(M), U_{k}(M ∖ {0})) ⟶ (C_{k}(M), C_{k}(M ∖ {0}))

is a homology equivalence. The space U_{k}(M) decomposes up to homeomorphism as D^{n} × C_{k−1}(M ∖ {0}) and similarly U_{k}(M ∖ {0}) ≅ (D^{n} ∖ {0}) × C_{k−1}(M ∖ {0}), so the left-hand side of (5.1) is homeomorphic to

(D^{n}, D^{n} ∖ {0}) ∧ C_{k−1}(M ∖ {0})_{+}.

Composing with the homotopy equivalence ∂D^{n} → D^{n} ∖ {0} we obtain the following diagram:

(5.2)  _{C_{k}(M ∖ {0}) C_{k}(M) (C_{k}(M), C_{k}(M ∖ {0})) S^{n−1} × C_{k−1}(M ∖ {0}) D^{n} × C_{k−1}(M ∖ {0}) Σ^{n}(C_{k−1}(M ∖ {0})_{+}) t_{k−1} (⋆)}

where the rows are cofibre sequences and the rightmost vertical map (⋆) is a homology equivalence. The map t_{k−1} may be described as radially expanding the configuration in C_{k−1}(M ∖ {0}) away from 0 until it has no points in D, and then adding the point in S^{n−1} = ∂D to the configuration.

Remark 5.3 Note that the bottom left horizontal map of (5.2) is homotopy split-surjective, so the maps

Σ^{n−1}(C_{k−1}(M ∖ {0})_{+}) −−→ S^{n−1} × C_{k−1}(M ∖ {0}) ⟶ D^{n} × C_{k−1}(M ∖ {0})

induce split short exact sequences on homology, corresponding to the Künneth decomposition for S^{n−1} × C_{k−1}(M ∖ {0}). The dotted arrow is the connecting homomorphism and only exists on homology.

The upshot of this discussion is the following lemma, where −−→ indicates a map which is only defined on homology.

Lemma 5.4 There are maps C_{k}(M ∖{0}) ⟶ C_{k}(M) −−→ Σ^{n}(C_{k−1}(M ∖{0})_{+}) which induce a long exact sequence on homology. The connecting homomorphism is the composite

Σ^{n−1}(C_{k−1}(M ∖ {0})_{+}) −−→ S^{n−1} × C_{k−1}(M ∖ {0}) ⟶ C_{k}(M ∖ {0}).

The first of these two maps is the inclusion of a direct summand of the homology of S^{n−1} × C_{k}(M ∖ {0}) and the second is the map t_{k−1} described above.

5.3. Configuration spaces on cylinders. For the remainder of this section n = dim(M) will always be assumed even.

Some natural homology classes. We will need to do some calculations inside the homology group H_{n−1}(C_{k}(ℝ^{n} ∖{0}); ℤ) of punctured Euclidean space. There are certain natural elements of this group which one can write down. For example we have the following elements (see also Figure 5.1):

(a) For any 0 ⩽ j ⩽ k − 1 we have a map Δ_{j}: S^{n−1} → C_{k}(ℝ^{n} ∖ {0}) which sends v ∈ S^{n−1} to the configuration {v, p_{1}, …, p_{k−1}}, where p_{1}, …, p_{k−1} are arbitrary fixed points in ℝ^{n} ∖ {0} with |p_{i}| < 1 for i ⩽ j and |p_{i}| > 1 for i > j. By abuse of notation we denote the element (Δ_{j})_{∗}([S^{n−1}]) simply by Δ_{j} ∈ H_{n−1}(C_{k}(ℝ^{n} ∖ {0}); ℤ). We will systematically use this abuse of notation for maps S^{n−1} → C_{k}(ℝ^{n} ∖ {0}).

(b) We also have a map π: ℝℙ^{n−1} → C_{k}(ℝ^{n} ∖ {0}) which sends {v, −v} ∈ ℝℙ^{n−1} = S^{n−1}/ ∼ to the configuration \{\underline{2} + v, \underline{2} − v, p_{1}, …, p_{k−2}\}, where \underline{2} = (2, 0, …, 0) and p_{1}, …, p_{k−2} are fixed points in ℝ^{n} ∖B_{1}(\underline{2}). This gives us an element π ∈ H_{n−1}(C_{k}(ℝ^{n} ∖ {0}); ℤ).

(c) Composing this map with the double covering S^{n−1} → ℝℙ^{n−1} gives a map representing 2π. This is homotopic to the map τ: S^{n−1} → C_{k}(ℝ^{n} ∖ {0}) which sends v ∈ S^{n−1} to the configuration {p_{1}, s(v), p_{2}, …, p_{k−1}}, where s: S^{n−1} → ℝ^{n} ∖{0} is an embedding so that p_{1} is in the interior of s(S^{d−1}) and 0, p_{2}, …, p_{k−1} are in its exterior.

(d) More generally, for any 1 ⩽ j ⩽ k − 1 we can define a map τ_{j}: S^{n−1} → ℝ^{n} ∖ {0} which sends v ∈ S^{n−1} to the configuration {p_{1}, …, p_{j}, s(v), p_{j+1}, …, p_{k−1}}, where p_{1}, …, p_{j} are in the interior of s(S^{n−1}) and 0, p_{j+1}, …, p_{k−1} are in its exterior. So τ_{1} = τ = 2π.

Relations between homology classes. Let P^{n} denote the closed ndimensional disc 𝔻^{n} with two open subdiscs (whose closures are disjoint) removed; this is the n-dimensional pair-of-pants. Consider the map r: P^{n} → C_{k}(ℝ^{n} ∖ {0}) pictured in Figure 5.2. The image r_{∗}([∂P^{n}]) of the fundamental class of its boundary is the class Δ_{j+1} − Δ_{j} − τ_{1}, which is therefore equal to zero in H_{n−1}(C_{k}(ℝ^{n} ∖ {0}); ℤ). Similarly the map r′: P^{n} → C_{k}(ℝ^{n} ∖ {0}) pictured in Figure 5.2 shows that τ_{j+1} − τ_{j} − τ_{1} = 0. Hence by induction and the fact that τ_{1} = 2π we have

(5.3)  Δ_{j} = Δ_{0} + 2jπ  and  τ_{j} = 2jπ.

\begin{matrix}Δ_{3} \\ π \\ τ_{2} \end{matrix}

Figure 5.1. Examples of homology classes in H_{1}(C_{6}(ℝ^{2} ∖ {0}); ℤ). The small circle denotes the puncture 0 and bullets denote points of the configuration.

. . . . . . r: . . . . . . r′:

Figure 5.2. Pictures of maps r, r′: P^{n} → C_{k}(ℝ^{n} ∖{0}) such that r_{∗}([∂P^{n}]) = Δ_{j+1} −Δ_{j} −τ_{1} and r′_{∗}([∂P^{n}]) = τ_{j+1} −τ_{j} −τ_{1}. In each case there are j + 1 fixed points in the bounded white region and k−j −2 fixed points in the unbounded white region. The remaining point is in the shaded region; its position is parametrised by P^{n}.

Now let Δ̂ denote the image of the fundamental class under the map S^{n−1} → C_{k}(ℝ^{n} ∖ {0}) which sends v to {v, 2v, …, kv}. This map can be homotoped to the map

(5.4)  S^{n−1} ⟶ S^{n−1} ∨ ⋯ ∨ S^{n−1} ⟶ C_{k}(ℝ^{n} ∖ {0})

which collapses k − 1 equators to get a wedge sum of k copies of S^{n−1} and then applies the maps Δ_{0}, …, Δ_{k−1} on these summands. To see this, note that the k−1 equators divide S^{n−1} into k slices (see Figure 5.3) and let q_{i}: S^{n−1} → S^{n−1} be the quotient map that collapses everything but the ith slice of the sphere to a point. Note that the map q_{{1,...,k}}: S^{n−1} → C_{k}(ℝ^{n} ∖ {0}) defined by sending v ∈ S^{n−1} to {q_{1}(v), 2q_{2}(v), …, kq_{k}(v)} is precisely the map (5.4) described above. On the other hand, each q_{i} is homotopic to the identity, and these homotopies assemble to give a homotopy from q_{{1,...,k}} to v ↦ {v, 2v, …, kv}. Hence Δ̂ = Δ_{0} + ⋯ + Δ_{k−1} and so by (5.3),

(5.5)  Δ̂ = kΔ_{0} + k(k − 1)π.

Figure 5.3. The slices of S^{n−1} when k = 5.

Similarly we let τ̂ denote the image of the fundamental class under the map S^{n−1} → C_{k}(ℝ^{n} ∖ {0}) which sends v to p_{1} + {0, v, 2v, …, (k − 1)v}, where p_{1} is a fixed point in ℝ^{n} with |p_{1}| ⩾ k. Just as above, we can homotope this to see that τ̂ = τ_{1} + ⋯ τ_{k−1} and so by (5.3),

(5.6)  τ̂ = k(k − 1)π.

One can see this very directly in the case n = 2. In this case we are talking about H_{1}(C_{k}(ℝ^{2}); ℤ) = β_{k}/[β_{k}, β_{k}] = ℤ{π}, where β_{k} denotes the braid group on k strands. Any one of the standard generators σ_{1}, …, σ_{k−1} of β_{k}, which interchange two consecutive strands, is sent to the generator π. The element τ̂ is the image of the full twist of all k strands, which can be written as a product of k(k − 1) generating elements, and so after abelianisation we have τ̂ = k(k − 1)π. We now apply the above discussion to prove the following:

Lemma 5.5 For any map f: S^{n−1} → S^{n−1} define σ_{f}: S^{n−1} → C_{k}(ℝ^{n} ∖ {0}) by sending v to \{v, v + \frac{1}{k} f(v), …, v + \frac{k−1}{k}f(v)\}. Denoting the image of the fundamental class under this map also by σ_{f} we have

(5.7)  σ_{f} = kΔ_{0} + deg(f)k(k − 1)π.

Proof. Note that if deg(f) = 1 then σ_{f} = σ_{id} = Δ̂ so this is just (5.5). In general this can be seen as follows. Write d = deg(f) and first assume that d > 0. Denote the constant map to the basepoint by ∗: S^{n−1} → S^{n−1} and the map S^{n−1} → S^{n−1} ∨ ⋯ ∨ S^{n−1} which collapses d − 1 equators by c_{d}. Then σ_{f} can be homotoped to the map

v ↦ \{s(v), s(v) + \frac{1}{k} g(v), …, s(v) + \frac{k−1}{k}g(v)\}

where s = (id + ∗ + ⋯ + ∗) ∘ c_{d} and g = (id + id + ⋯ + id) ∘ c_{d}, which is in turn homotopic to the map (Δ̂ + τ̂ + ⋯ + τ̂) ∘ c_{d}: S^{n−1} → C_{k}(ℝ^{n} ∖ {0}). Therefore the homology class σ_{f} is equal to Δ̂ + (d − 1)τ̂, which is the claimed formula by (5.5) and (5.6). If d ⩽ 0 we can instead take s = (id + ∗ + ⋯ + ∗) ∘ c_{2−d} and g = (id + r + ⋯ + r) ∘ c_{2−d}, where r is a reflection of S^{n−1}, to see that σ_{f} is homotopic to the map (Δ̂ + τ̂ ∘ r + ⋯ + τ̂ ∘ r) ∘ c_{2−d}. The image of the fundamental class [S^{n−1}] under τ̂ ∘ r is just −τ̂, so we again get that the homology class σ_{f} is equal to Δ̂ + (d − 1)τ̂. □

Remark 5.6 Rationally, the (n − 1)st homology of C_{k}(ℝ^{n} ∖ {0}) is known to be two-dimensional by the presentation of the bigraded ℚ-algebra H_{∗}(C_{∗}(ℝ^{n} ∖ {0}); ℚ) given in Proposition 3.4 of [Randal-Williams, 2013b]. Specifically, it is generated by the elements Δ_{0} and Δ_{1}, corresponding to [k − 1] · Δ and [k − 2] · Δ · [1] in the notation of the cited paper.

5.4. Proof of Theorem D. Abbreviate C_{l}(M ∖ {0}) by just C_{l}. Fix a field 𝔽 of characteristic p > 0 and write H̃_{∗}(−) = H̃_{∗}(−; 𝔽). Recall from Lemma 5.4 that we have a long exact sequence on homology

⋯ ⟶ H̃_{∗}(Σ^{n−1}((C_{k−1})_{+})) ⟶ H̃_{∗}(C_{k}) ⟶ H̃_{∗}(C_{k}(M)) ⟶ ⋯

and denote the left-hand map above by T_{k,∗}. By exactness we have:

dim(H̃_{∗}(C_{k}(M))) = dim(codomain(T_{k,∗})) + dim(domain(T_{k,∗−1}))
− rank(T_{k,∗}) − rank(T_{k,∗−1}).

(5.8)

Hence in order to identify H̃_{∗}(C_{k}(M)) and H̃_{∗}(C_{rk}(M)) in a range it suffices to identify the linear maps T_{k,∗} and T_{rk,∗} in a range.

Proof of Theorem D. Fix a positive integer r ⩾ 2 coprime to p. We will construct maps a, b and c such that the square

(5.9)  \begin{matrix}S^{n−1} × C_{k−1} \\ S^{n−1} × C_{rk−r} \\ S^{n−1} × C_{rk−1} \\ C_{k} \\ C_{rk} \\ a \\ b \\ c \\ t_{k−1} \\ t_{rk−1} \end{matrix}

commutes on homology with coefficients in 𝔽. Applying H̃_{∗}(−) and restricting to a direct summand (see Remark 5.3) on the top row gives a commutative square

(5.10)  \begin{matrix}H̃_{∗+1}(Σ^{n}(C_{k−1})_{+}) \\ H̃_{∗+1}(Σ^{n}(C_{rk−r})_{+}) \\ H̃_{∗+1}(Σ^{n}(C_{rk−1})_{+}) \\ H̃_{∗}(C_{k}) \\ H̃_{∗}(C_{rk}). \\ α \\ β \\ c_{∗} \\ T_{k,∗} \\ T_{rk,∗} \end{matrix}

Throughout this section we will abbreviate μ = μ[M ∖{0}] and ν = ν[M ∖{0}]. Recall that we defined the function

λ(k) = λ[M](k) = min{ν(k), ν(k − 1) + n − 1, μ(rk − i) | i = 2, …, r},

where n is the dimension of M. We will show that α, β and c_{∗} are isomorphisms in the range ∗ ⩽ λ(k), therefore identifying the maps T_{k,∗} and T_{rk,∗} in this range. Hence by (5.8) the vector spaces H̃_{∗}(C_{k}(M)) and H̃_{∗}(C_{rk}(M)) have the same dimension for ∗ ⩽ λ(k), which is Theorem D.

Remark 5.7 The first two terms of λ(k) come from our use of the replication map and Theorem B, which tells us that the r-replication map induces isomorphisms in the stable range ν. The remaining terms come from our use of the classical stabilisation map, which by definition induces isomorphisms in the range μ. If we assume that μ is non-decreasing (so ν = μ) and r, k ⩾ 2 then the range ∗ ⩽ λ(k) simplifies to

∗ ⩽ min{μ(k), μ(k − 1) + n − 1}.

For example if μ(k) = ak + b then this is

∗ ⩽ ak + b  if n ⩾ a + 1
∗ ⩽ ak + b − (a + 1 − n)  if n < a + 1,

i.e. the same as the stable range, except possibly shifted down by a constant if the manifold is low-dimensional compared to the slope of the stable range.

Constructing the maps. Fix a basepoint 0 ∈ M. By Lemma 5.2 we can choose a vector field v on M which is non-vanishing except possibly at 0. This has an associated one-parameter family of diffeomorphisms ϕ_{t}. Define the rreplication map

ρ_{r,k}: C_{k}(M ∖ {0}) ⟶ C_{rk}(M ∖ {0})

to take a configuration c = {x_{1}, …, x_{k}} to the configuration

{ϕ_{it/r}(x_{1}), …, ϕ_{it/r}(x_{k}) | 0 ⩽ i ⩽ r − 1},

where t = t(c) > 0 is sufficiently small that ϕ_{s}(x_{i}) ≠ ϕ_{u}(x_{j}) for s, u ∈ (0, t) unless i = j and s = u. This agrees up to homotopy with the earlier definition of the r-replication map under the identifications C_{k}^{δ}(M ∖ {0}) ≃ C_{k}(M ∖ {0}) and C_{rk}^{δ}(M ∖ {0}) ≃ C_{rk}(M ∖ {0}). We now define

a = id × ρ_{r,k−1}
b = (pr_{1}, t_{rk−2}) ∘ ⋯ ∘ (pr_{1}, t_{rk−r})
c = ρ_{r,k}.

In other words a and c replace each point of the configuration by r copies in the direction determined by the vector field, whereas b adds r − 1 new points near the missing point 0 in the direction determined by the vector in S^{n−1}.

Isomorphisms in a range. The vector field is non-vanishing on M ∖ {0}, so Theorem B tells us that the r-replication map ρ_{r,k} induces isomorphisms in the stable range on homology with ℤ_{(p)} coefficients, and hence also with 𝔽 coefficients. Hence c_{∗} is an isomorphism in the stable range ∗ ⩽ ν(k). The map ρ_{r,k−1} induces isomorphisms on 𝔽-homology up to degree ν(k − 1), so its suspension Σ^{n}((ρ_{r,k−1})_{+}) induces isomorphisms up to degree ν(k − 1) + n.

The map that this induces on H̃_{∗+1}(−) is α, which is therefore an isomorphism in the range ∗ ⩽ ν(k − 1) + n − 1. For the map β consider the map of (trivial) fibre bundles

S^{n−1} × C_{rk−i} S^{n−1} × C_{rk−i+1} S^{n−1} (pr_{1}, t_{rk−i})

for i = 2, …, r. Its fibre over a point in S^{n−1} is the classical stabilisation map and therefore induces isomorphisms on 𝔽-homology up to degree μ(rk − i). Hence by the relative Serre spectral sequence the map (pr_{1}, t_{rk−i}) also induces isomorphisms on 𝔽-homology in this range. So the map b induces isomorphisms on 𝔽-homology up to degree min{μ(rk − i) | 2 ⩽ i ⩽ r}. In general, for a map f: S^{d} × A → S^{d} × B over S^{d}, the map on homology under the Künneth isomorphism, f_{∗}: H_{∗}(A) ⊕ H_{∗−d}(A) → H_{∗}(B) ⊕ H_{∗−d}(B), is triangular – more precisely the component H_{∗}(A) → H_{∗−d}(B) is zero. To see this note that a representing cycle c for an element in the H_{∗}(A) component can be taken to have support in a single fibre. Since f is a map over S^{d} the image f_{♯}(c) will also have support in a single fibre, and therefore the image f_{∗}([c]) will be in the H_{∗}(B) component of the Künneth decomposition of the right-hand side. Hence if f induces an isomorphism on homology, it also restricts to isomorphisms between each of the direct summands in the Künneth decompositions of the source and target. Applying this fact to the map b we obtain that β is an isomorphism in the range ∗ ⩽ min{μ(rk − i) | 2 ⩽ i ⩽ r}. Hence each of α, β and c_{∗} are isomorphisms in the range ∗ ⩽ λ(k).

Commutativity. It therefore remains to show that the square (5.9) commutes on 𝔽-homology. Choose a coordinate neighbourhood U ≅ ℝ^{n} of 0 ∈ M and define the map

(5.11)  ζ: C_{r}(ℝ^{n} ∖ {0}) × C_{k−1}(M ∖ {0}) ⟶ C_{rk}(M ∖ {0})

to first apply the map ρ_{r,k−1} to the configuration in M ∖ {0}, i.e. replace each point by r copies according to the vector field, then push the resulting configuration radially away from 0 so that it is disjoint from U, and finally insert the configuration of r points in ℝ^{n} ∖ {0} = U ∖ {0} into the vacated space. Choosing a trivialisation of T M over U ≅ ℝ^{n}, the vector field v restricts to a map ℝ^{n} → ℝ^{n} which is non-vanishing on S^{n−1}, so we may rescale it to obtain a map f: S^{n−1} → S^{n−1}. Recall from Lemma 5.5 that such a map induces a map σ_{f}: S^{n−1} → C_{r}(ℝ^{n} ∖ {0}). One can then easily see that the two ways c ∘ t_{k−1} and t_{rk−1} ∘ b ∘ a around the square (5.9) are homotopic to

ζ ∘ (σ_{f} × id) and ζ ∘ (σ_{id} × id): S^{n−1} × C_{k−1}(M ∖ {0}) ⟶ C_{rk}(M ∖ {0})

respectively. It suffices to show that σ_{f} and σ_{id} : S^{n−1} → C_{r}(ℝ^{n} ∖ {0}) induce the same map on 𝔽-homology, and we only need to check this on the fundamental class. Using our abuse of notation from §5.3 this means that we just need to check that the homology classes σ_{f} and σ_{id} in H_{n−1}(C_{r}(ℝ^{n} ∖ {0}); 𝔽) are equal. The degree of f: S^{n−1} → S^{n−1} is χ by the Poincaré-Hopf theorem (c.f. Definition 5.1) so by Lemma 5.5 we have

σ_{f} = rΔ_{0} + χr(r − 1)π
σ_{id} = rΔ_{0} + r(r − 1)π

in H_{n−1}(C_{r}(ℝ^{n}∖{0}); ℤ). Their difference is (χ−1)r(r−1)π, which is divisible by p = char(𝔽) by hypothesis. Hence the difference σ_{f} − σ_{id} is indeed zero in H_{n−1}(C_{r}(ℝ^{n} ∖ {0}); 𝔽), and so the square (5.9) commutes on 𝔽-homology. □

5.5. The case of the two-sphere. For M = S^{2} we have the well-known calculation H_{1}(C_{k}(S^{2}); ℤ) ≅ ℤ/(2k − 2)ℤ for k ⩾ 2 obtained from a presentation for π_{1}(C_{k}(S^{2})) (see [Fadell and Van Buskirk, 1962]). The degree-one 𝔽_{p}- homology is therefore either one- or zero-dimensional, depending on whether p | 2k − 2 or not. So the statement of Theorem D in degree 1 for M = S^{2} for mod-p coefficients reduces to the following purely number-theoretic statement: if p is a prime and r is a positive integer such that p | r − 1 then p | 2k − 2 if and only if p | 2rk − 2. This is of course obviously true: we have r − 1 = ap for some a, so 2rk − 2 = 2k + 2kap − 2 ≡ 2k − 2 (mod p).

5.6. Generalisation to configurations with labels in a bundle. Theorem D generalises directly to configuration spaces C_{k}(M; θ) with labels in a bundle θ: E → M with path-connected fibres.

Theorem D′ Let M be a closed, connected, smooth manifold with Euler characteristic χ and let θ: E → M be a fibre bundle with path-connected fibres. Choose a field 𝔽 of positive characteristic p and let r ⩾ 2 be an integer coprime to p such that p divides (χ − 1)(r − 1). Then there are isomorphisms

H_{∗}(C_{k}(M; θ); 𝔽) ≅ H_{∗}(C_{rk}(M; θ); 𝔽)

in the range ∗ ⩽ λ[M; θ](k).

The function λ[M; θ] is defined just as λ[M], namely:

λ[M; θ](k) = min{ν(k), ν(k − 1) + n − 1, μ(rk − i) | i = 2, …, r},

where μ = μ[M^{⋆}; θ^{⋆}], ν = ν[M^{⋆}; θ^{⋆}] and M^{⋆}, θ^{⋆} denote M ∖ {∗} and θ|_{M∖{∗}} respectively. These two functions are defined analogously to Definition 1.1, using the stabilisation and scanning maps for configuration spaces with labels in a bundle. In the remainder of this subsection we sketch how to generalise the proof of Theorem D to a proof of Theorem D′. The proof follows the same steps. In §5.2 one has to additionally choose a trivialisation of θ over the embedded disc D ⊆ M, and analogously to Lemma 5.4 there is a cofibre sequence

C_{k}(M^{⋆}; θ^{⋆}) ⟶ C_{k}(M; θ) −−→ Σ^{n}((F × C_{k−1}(M^{⋆}; θ^{⋆}))_{+}),

where F is the typical fibre of θ. The description of the connecting homomorphism for the long exact sequence on homology is exactly analogous, using the trivialisation of θ over D to determine the label of the new point which is added to the configuration near 0 ∈ D. In the diagram (5.9) the top three spaces are replaced by their cartesian products with F. The maps c_{∗} and α are isomorphisms in the range ∗ ⩽ λ(k) for the same reasons as before, using Theorem B′ instead of Theorem B. The map b is a composition of maps of fibre bundles over F ×S^{n−1} and the maps of fibres are classical stabilisation maps for configuration spaces with labels in a bundle, and so are isomorphisms on homology in the stable range for the stabilisation map (c.f. Proposition B.3 and the appendix of [Kupers and Miller, 2014a]). The rest of the argument that β is an isomorphism in the range ∗ ⩽ λ(k) goes through as before. For commutativity: the map ζ can be defined similarly, using the chosen trivialisation of θ over D. The input is now a configuration of k−1 points in M ∖{0} with labels in θ and a configuration of r points in ℝ^{n}∖{0} with labels in the trivial bundle with fibre F, and the output is a configuration of rk points in M ∖{0} with labels in θ. The map f: S^{n−1} → S^{n−1}, corresponding to the restriction of the vector field to ∂D, induces a map σ_{f}: F × S^{n−1} → C_{r}(ℝ^{n} ∖ {0}; F). The two ways around the square (5.9) are homotopic to ζ ∘(σ_{f} ×id) and ζ ∘(σ_{id}×id). Hence we just need to show that σ_{f} and σ_{id} : F × S^{n−1} → C_{r}(ℝ^{n} ∖ {0}; F) induce the same map on 𝔽-homology. Now as in §5.3 we need to find formulas, in terms of more basic classes, for (σ_{f})_{∗}(x), for any class x ∈ H_{∗}(F × S^{n−1}). Previously we showed that when F = ∗ and x = [S^{n−1}] we have

(σ_{f})_{∗}([S^{n−1}]) = rΔ_{0} + deg(f)r(r − 1)π ∈ H_{n−1}(C_{r}(ℝ^{n} ∖ {0}); ℤ).

By the Künneth decomposition H_{∗}(F ×S^{n−1}) = H_{∗}(F)⊕H_{∗−n+1}(F) it suffices to show that (σ_{f})_{∗}(x × [∗]) − (σ_{id})_{∗}(x × [∗]) and (σ_{f})_{∗}(x × [S^{n−1}]) − (σ_{id})_{∗}(x × [S^{n−1}]) are zero on 𝔽-homology for any class x ∈ H_{∗}(F). It is easy to see that in fact

(σ_{f})_{∗}(x × [∗]) = (σ_{id})_{∗}(x × [∗]) ∈ H_{∗}(C_{r}(ℝ^{n} ∖ {0}; F); ℤ)

and therefore also on 𝔽-homology. One can define classes π(x), Δ_{i}(x), τ_{i}(x) etc. in H_{∗+n−1}(C_{r}(ℝ^{n} ∖ {0}; F); ℤ) just as in §5.3 and by the same arguments as before we have

(σ_{f})_{∗}(x×[S^{n−1}]) = rΔ_{0}(x)+deg(f)r(r−1)π(x) ∈ H_{∗+n−1}(C_{r}(ℝ^{n}∖{0}; F); ℤ).

Hence (σ_{f})_{∗}(x × [S^{n−1}]) − (σ_{id})_{∗}(x × [S^{n−1}]) is equal to (deg(f) − 1)r(r − 1)π(x) = (χ − 1)r(r − 1)π(x) and is therefore zero on 𝔽-homology since p divides (χ − 1)(r − 1). This completes the sketch of the proof of Theorem D′.

5.7. Combining Theorems A and D. We now prove Corollary E, concerning the homology of configuration spaces on even-dimensional manifolds with coefficients in a field of odd characteristic. In fact, our methods also partially recover the known homological stability results for odd-dimensional manifolds and for fields of characteristic 2 or 0. The complete statement of what may be deduced by combining Theorems A and D is as follows:

Corollary 5.8 Let M be a closed, connected, smooth manifold with Euler characteristic χ and let 𝔽 be a field of characteristic p. Then, in the stable range (resp. the range ∗ ⩽ λ(k) for lines 4–9 ), the homology group H_{∗}(C_{k}(M); 𝔽) depends only on the quantity stated in Table 5.1.

dim.  conditions  H_{∗}(C_{k}(M); 𝔽) depends only on  ♯
odd p ≠ 2 —  —  1  A
p = 2 —  parity of k  2  A
even p = 0 —  whether 2k = χ  1 or 2^{∗} A
p odd —  min{(2k − χ)_{p}, (χ)_{p} + 1}  (χ)_{p} + 2 A,D
χ ≢ 0 mod p whether p divides 2k − χ  2  A,D
χ ≡ 1 mod p —  1  A,D
p = 2 —  (k)_{2}  ∞  D
(χ)_{2} ⩾ 1  min{(k)_{2}, (χ)_{2}}  (χ)_{2} + 1 A,D
(χ)_{2} = 1  parity of k  2  A,D

Table 5.1. The second-from-right column is the maximum number of different values of H_{∗}(C_{k}(M); 𝔽) in the stable range (resp. the range ∗ ⩽ λ(k) in lines 4–9). The rightmost column indicates which theorem(s) each line follows from.

∗ There is only one “stable homology” on the third line, although when χ is even there may be a single exception in the stable range when k = \frac{χ}{2} .

Proof. The first two lines follow directly from the odd part of Theorem A, noting that a map of spaces which induces an isomorphism with ℤ[\frac{1}{2}] coefficients also induces isomorphisms with coefficients in any field of characteristic different from 2. The third line follows directly from the even part of Theorem A, since a map of spaces inducing isomorphisms on homology with ℤ_{(p)} coefficients also induces isomorphisms with ℚ or 𝔽_{p} coefficients, and therefore with coefficients in any field of characteristic 0 or p. (Given j, k in the stable range and not equal to \frac{χ}{2}, choose any prime p which divides neither 2j − χ nor 2k − χ and apply Theorem A to get an isomorphism with ℤ_{(p)} coefficients.) For the fourth line there are two cases to consider. For the first case suppose that (2j −χ)_{p} = (2k −χ)_{p} ⩽ (χ)_{p}. In this case the result follows from Theorem A. Now suppose that (2j − χ)_{p}, (2k − χ)_{p} > (χ)_{p} and write x′ for x/p^{(x)_{p}} for each number x. The assumption implies that (j)_{p} = (χ)_{p} = (k)_{p} and that p divides both 2k′ − χ′ and 2j′ − χ′, so that j′ ≡ k′ ≢ 0 mod p. Hence we may choose l such that lk′ ≡ 1 mod p, so that by Theorem D we have:

p|(lj′ − 1) so H_{∗}(C_{k}(M); 𝔽_{p}) ≅ H_{∗}(C_{lj′k}(M); 𝔽_{p}) for ∗ ⩽ min(λ(k), λ(lj′k)),
p|(lk′ − 1) so H_{∗}(C_{j}(M); 𝔽_{p}) ≅ H_{∗}(C_{lk′j}(M); 𝔽_{p}) for ∗ ⩽ min(λ(j), λ(lk′j)).

Since lj′k = lk′j we have the required isomorphism in the intersection of these two ranges. Note that l may be chosen arbitrarily large and λ is divergent, so this is precisely the range ∗ ⩽ min(λ(k), λ(j)). The fifth line is the special case of the fourth line when (χ)_{p} = 0. Sixth line: Suppose that ∗ ⩽ min(λ(k), λ(j)); we will consider three cases. First, if j and k are both in pℤ then (2j − χ)_{p} = 0 = (2k − χ)_{p}, so we have an isomorphism H_{∗}(C_{j}(M); 𝔽) ≅ H_{∗}(C_{k}(M); 𝔽) by Theorem A. Second, if j and k are both not in pℤ then (j)_{p} = 0 = (k)_{p} and the isomorphism follows from Theorem D (since p | χ − 1, we may take r in Theorem D to be any integer coprime to p). Finally, suppose that j ∈ pℤ and k ∉ pℤ. Then

(2j − χ)_{p} = 0 = (2(jkl + χ) − χ)_{p}  and  (jkl + χ)_{p} = 0 = (k)_{p}

for any l. Since λ diverges and l may be chosen arbitrarily large we have an isomorphism H_{∗}(C_{j}(M); 𝔽) ≅ H_{∗}(C_{k}(M); 𝔽) by Theorems A and D. The seventh line follows directly from Theorem D: when p = 2 we may take r to be any odd integer, so there are isomorphisms in the range ∗ ⩽ min(λ(j), λ(k)) between any j, k with the same 2-adic valuation. To deduce the eighth line from the seventh we need to show that there are isomorphisms in this range whenever (j)_{2} and (k)_{2} are both at least (χ)_{2}. In this case we have (2k − χ)_{2} = (χ)_{2} = (2j − χ)_{2}, and so we can apply Theorem A (since we assumed that χ is even). Finally, the ninth line is a special case of the eighth line. □

Appendix A. The stability range for the torsion in configuration spaces

In this appendix we show that the stable range for homological stability of unordered configuration spaces may be improved to have slope 1 when taking ℤ[\frac{1}{2}] coefficients. We note that this has also recently been proved by a different method by [Kupers and Miller, 2014b]. We begin by proving this in a larger range when M = ℝ^{n} using Salvatore’s description [Salvatore, 2004] of Cohen’s calculations [Cohen et al., 1976], and then use this to deduce the slope 1 statement for general open, connected manifolds M. Our method for this second step is a slight variation of an argument due to Oscar Randal-Williams in [Randal-Williams, 2013a, §8]. From Salvatore’s description [Salvatore, 2004, page 537] of the homology of C(ℝ^{n}) (based on [Cohen et al., 1976, page 227]), we obtain the following. A non-empty sequence of positive integers I = (i_{1}, …, i_{ℓ(I)}) with ℓ(I) ⩾ 0 is said to be (n, p)-admissible if it is weakly monotone and strictly bounded above by n. If p is odd, an admissible function for I is a function ϵ: {1, …, ℓ(I)} → {0, 1} satisfying

ϵ(j) ≡ i_{j} + i_{j−1} mod 2  for 2 ⩽ j ⩽ ℓ(I).

Observe that ϵ is determined by I and ϵ(1). If p = 2, define ϵ to be constant with value 0.

If p = 2, then H_{∗}(C(ℝ^{n}); 𝔽_{2}) is isomorphic to the free commutative graded algebra generated by the symbols Q_{ϵ(1),I}(ι) where I is an (n, p)-admissible sequence and ϵ is an admissible function. If p is odd, then H_{∗}(C(ℝ^{n}); 𝔽_{p}) is isomorphic to the free commutative graded algebra generated by the symbols Q_{ϵ(1),I}(ι) where I is an (n, p)-admissible sequence with i_{ℓ(I)} even and ϵ is an admissible function for I, and also (if n is even) the symbols Q_{ϵ(1),I}([ι, ι]) where I is an (n, p)-admissible sequence with i_{ℓ(I)} odd and ϵ is an admissible function for I. The homological degrees of ι := Q_{∅,∅}(ι) and [ι, ι] := Q_{∅,∅}([ι, ι]) are 0 and n−1, and the configuration degrees are 1 and 2. The homological and configuration degrees of the other generators are

h(Q_{ϵ(1),(i_{1},...,i_{k})}(α)) = ph(Q_{ϵ(2),(i_{2},...,i_{k})}(α)) + i_{1}(p − 1) − ϵ(1)
ν(Q_{ϵ(1),(i_{1},...,i_{k})}(α)) = pν(Q_{ϵ(2),(i_{2},...,i_{k})}(α)),

where α = ι or [ι, ι]. The degrees of a product of generators are:

h(xy) = h(x) + h(y),  ν(xy) = ν(x) + ν(y).

Multiplication by the class ι raises the configuration degree by 1 and hence defines a homomorphism

H_{∗}(C_{k−1}(ℝ^{n})) ⟶ H_{∗}(C_{k}(ℝ^{n})),

which is the same as that induced by the stabilisation map. We say that a class in H_{∗}(C_{k}(ℝ^{n}); 𝔽_{p}) is p-inceptive if it is not in the image of the stabilisation map C_{k−1}(ℝ^{n}) → C_{k}(ℝ^{n}) on mod-p homology. By the above a class is p-inceptive if and only if it is not in the principal ideal generated by ι.

Lemma A.1 In H_{∗}(C_{k}(ℝ^{n}); 𝔽_{p}), the first p-inceptive class in a fixed configuration degree k is given in Table A.1 on page 791, where a = ⌊k/p⌋ and m_{p}(k) is the remainder after dividing k by p. Any case not covered in the table has no p-inceptive classes. Hence, by the above discussion, the stabilisation map C_{k−1}(ℝ^{n}) → C_{k}(ℝ^{n}) induces an isomorphism on H_{∗}(−; 𝔽_{p}) for smaller homological degrees.

p  n  k  class  homological degree
even  all  even  Q_{0,(1)}(ι)^{a}  a
odd  odd  ∈ pℤ  Q_{1,(2)}(ι)^{a}  a(2(p − 1) − 1)
\frac{odd}{3, 5}  \frac{⩾ 6, even}{4}  ⩾ p, odd  Q_{1,(2)}(ι)^{a}[ι, ι]^{m_{p}(k)/2}  a(2(p − 1) − 1) + (n − 1)m_{p}(k)/2

\frac{odd}{3, 5}  \frac{⩾ 6, even}{4}  even  Q_{1,(2)}(ι)^{a−1}[ι, ι]^{(p+m_{p}(k))/2} (a − 1)(2(p − 1) − 1) + (n − 1)(p + m_{p}(k))/2

≠ 2, 3, 5  4  \frac{even}{⩾ p, odd}  Q_{1,(2)}(ι)^{m_{2}(k)}[ι, ι]^{⌊k/2⌋}  m_{2}(k)(2(p − 1) − 1) + (n − 1)⌊k/2⌋
odd  2  even  [ι, ι]^{k/2}  k/2

Table A.1. The first p-inceptive class in degree k. Here, we write a = ⌊k/p⌋ and m_{p}(k) = p(k/p − ⌊k/p⌋) for the remainder after dividing k by p. Any case not covered in this table has no p-inceptive classes.

Proof. First observe that

h(Q_{ϵ(1),(i_{1},...,i_{k})}(ι)) ⩾ h(Q_{ϵ(i_{2}),(i_{2},...,i_{k})}(ι)^{p})
ν(Q_{ϵ(1),(i_{1},...,i_{k})}(ι)) = ν(Q_{ϵ(i_{2}),(i_{2},...,i_{k})}(ι)^{p})
h(Q_{ϵ(1),(i_{1},...,i_{k})}([ι, ι])) ⩾ h(Q_{ϵ(i_{2}),(i_{2},...,i_{k})}([ι, ι])^{p})
ν(Q_{ϵ(1),(i_{1},...,i_{k})}([ι, ι])) = ν(Q_{ϵ(i_{2}),(i_{2},...,i_{k})}([ι, ι])^{p})
h(Q_{1,(i)}(ι)) ⩽ h(Q_{ϵ,(j)}(ι))
ν(Q_{1,(i)}(ι)) = ν(Q_{ϵ,(j)}(ι))

where i ⩽ j in the bottom two rows. Hence the lowest p-inceptive class in a fixed configuration degree is a product whose factors are

Q_{1}(ι)  if p = 2
Q_{1,(2)}(ι)  if p odd and n odd
Q_{1,(2)}(ι), [ι, ι] if p odd and n even
[ι, ι]  if p is odd and n = 2.

This is enough to deduce the first two rows of the table, as well as the sixth. Second observe that, if p is odd and n is even, the first configuration degree in which a power of Q_{1,(2)}(ι) and a power of [ι, ι] both live is 2p, where ν(Q_{1,(2)}(ι)^{2}) = ν([ι, ι]^{p}), and

h(Q_{1,(2)}(ι)^{2}) = 4p − 6 < p(n − 1) = h([ι, ι]^{p}) ⇔ n ⩾ 6 or n = 4, p = 3, 5,

from which the third, fourth and fifth rows of the table follow. □

Lemma A.1 tells us in particular that for odd primes p the stabilisation map C_{k}(ℝ^{n}) → C_{k+1}(ℝ^{n}) induces an isomorphism on homology with 𝔽_{p} coefficients in the range ∗ ⩽ k. We now show that this implies that the same is true for the stabilisation map C_{k}(M) → C_{k+1}(M) for any smooth, connected, open manifold M of dimension at least 3. Our method for this is a slight variation of an argument due to Oscar Randal-Williams in [Randal-Williams, 2013a, §8].

Proposition A.2 Let M be a smooth, connected, open n-manifold with n ⩾ 3 and let A be an abelian group. If the stabilisation map on A-homology

H_{∗}(C_{k}(ℝ^{n}); A) ⟶ H_{∗}(C_{k+1}(ℝ^{n}); A)

is an isomorphism in the range ∗ ⩽ k then so is the stabilisation map on A-homology

H_{∗}(C_{k}(M); A) ⟶ H_{∗}(C_{k+1}(M); A).

So by Lemma A.1 the stabilisation map C_{k}(M) → C_{k+1}(M) induces isomorphisms on homology with 𝔽_{p} coefficients in the range ∗ ⩽ k for any odd prime p.

This result has also been recently proved by [Kupers and Miller, 2014b] using a different method along the lines of [Segal, 1979].

Proof. We will just write H_{∗}(−) for H_{∗}(−; A). Define R_{k}(M) to be the homotopy cofibre of the stabilisation map C_{k}(M) → C_{k+1}(M). Now the stabilisation map C_{k}(M) → C_{k+1}(M) is split-injective on homology (see [McDuff, 1975, page 103]) so it induces an isomorphism on homology in degree ∗ if and only if H̃_{∗}(R_{k}(M)) = 0. So the hypothesis of the proposition says that H̃_{∗}(R_{k}(ℝ^{n})) = 0 for ∗ ⩽ k and we would like to show that H̃_{∗}(R_{k}(M)) = 0 for ∗ ⩽ k. We refer to [Randal-Williams, 2013a] for background and any details which we omit in this proof – the line of argument is very similar. The proof is by induction on k. The base case k = 0 is obvious so we now fix k ⩾ 1 for the inductive step. For i ⩾ 0 let C_{l}(M)^{i} be the space of l-point subsets c of M together with an injection {0, …, i} → c. These fit together to form an semi-simplicial space C_{l}(M)^{•} augmented by C_{l}(M). The stabilisation map lifts to a map C_{l}(M)^{•} → C_{l+1}(M)^{•} of augmented semi-simplicial spaces. There is a fibre bundle π: C_{l}(M)^{i} → C̃_{i+1}(M), where C̃ denotes the ordered configuration space, given by sending an injection {0, …, i} → c to its image and remembering the induced ordering. Its fibre over a point is homeomorphic to C_{l−i−1}(M_{i+1}), where M_{i+1} denotes the manifold M with i + 1 points removed. Moreover the projection π commutes with the stabilisation map C_{l}(M)^{i} → C_{l+1}(M)^{i} and the map of fibres over a point is the stabilisation map C_{l−i−1}(M_{i+1}) → C_{l−i}(M_{i+1}). Any map of Serre fibrations over a fixed base space has an associated relative Serre spectral sequence; in this case it has second page

iẼ_{s,t}^{2} ≅ H_{s}(C̃_{i+1}(M); H̃_{t}(R_{l−i−1}(M_{i+1})))

and converges to H̃_{∗}(R_{l}(M)^{i}), where R_{l}(M)^{i} denotes the homotopy cofibre of the lift C_{l}(M)^{i} → C_{l+1}(M)^{i} of the stabilisation map. For 1 ⩽ j ⩽ k there are maps of augmented semi-simplicial spaces C_{k−j}(M) × C_{j}(ℝ^{n})^{•} → C_{k}(M)^{•} defined similarly to the stabilisation map, except one stabilises by adding the given configuration in ℝ^{n} instead of just a single point. In [Randal-Williams, 2013a, §8] it is explained how these induce maps of semisimplicial spaces R_{k−j−1}(M) ∧ R_{j}(ℝ^{n})^{•} → R_{k}(M)^{•} for 1 ⩽ j ⩽ k. Note that when j = k we have R_{−1}(M) = S^{0} and this is just the map R_{k}(ℝ^{n})^{•} → R_{k}(M)^{•} induced by an embedding ℝ^{n} ↪ M. Each semi-simplicial space has an associated spectral sequence so we obtain a map ^{j}Ē → E of spectral sequences whose first pages are

jĒ_{s,t}^{1} ≅ H̃_{t}(R_{k−j−1}(M) ∧ R_{j}(ℝ^{n})^{s})
E_{s,t}^{1} ≅ H̃_{t}(R_{k}(M)^{s}).

Note that these are first quadrant plus an extra column {s = −1, t ⩾ 0}. The spectral sequence E converges to H̃_{∗+1} of the homotopy cofibre of the map ∥R_{k}(M)^{•}∥ → R_{k}(M) induced by the augmentation map. Since taking homotopy cofibres commutes with taking geometric realisation of semi-simplicial spaces this space can also be obtained as follows: first take the homotopy cofibres of the maps ∥C_{k}(M)^{•}∥ → C_{k}(M) and ∥C_{k+1}(M)^{•}∥ → C_{k+1}(M); these are related by a map induced by stabilisation; then take the homotopy cofibre of this map. Now the augmented semi-simplicial space C_{k}(M)^{•} is a (k − 1)-resolution [Randal-Williams, 2013a, Proposition 6.1], i.e. the map ∥C_{k}(M)^{•}∥ → C_{k}(M) is (k − 1)-connected. Hence the spectral sequence E converges to zero in total degree ∗ ⩽ k − 1. The inductive hypothesis says that

(IH)  H̃_{∗}(R_{l}(M)) = 0 for ∗ ⩽ l < k

and the hypothesis of the proposition says that

(Hyp)  H̃_{∗}(R_{l}(ℝ^{n})) = 0 for ∗ ⩽ l.

From (IH) we deduce that ^{i}Ẽ_{s,t}^{2} = 0 for t ⩽ l − i − 1 so the spectral sequence iẼ converges to zero in total degree ∗ ⩽ l − i − 1, so

(A.1)  H̃_{∗}(R_{l}(M)^{i}) = 0 for ∗ ⩽ l − i − 1 and i ⩾ 0.

In other words:

(A.2)  E_{s,t}^{1} = 0 for t ⩽ k − s − 1 and s ⩾ 0.

Also, using the Künneth theorem, (A.1) and (IH) we deduce that

(A.3)  ^{j}Ē_{s,t}^{1} = 0 for t ⩽ k − s − 1,

where for the case {s = −1 and j = k} we also need to use (Hyp). We now make the following:

Claim For 1 ⩽ j ⩽ k the map ^{j}Ē_{j,k−j}^{1} → E_{j,k−j}^{1} is surjective.

The verification of this claim is delayed until the end of the proof. Now a diagram chase in the following:

jĒ_{j,k−j}^{1}^{j}Ē_{j,k−j}^{j+1}^{j}Ē_{−1,k}^{j+1}^{j}Ē_{−1,k}^{1} E_{j,k−j}^{1} E_{j,k−j}^{j+1} E_{−1,k}^{j+1} E_{−1,k}^{1} = 0 d̄_{j+1} d_{j+1}

shows that the differential d_{j+1} : E_{j,k−j}^{j+1} → E_{−1,k}^{j+1} is zero for 1 ⩽ j ⩽ k. Now we can deduce that the first differential d_{1} : E_{0,t}^{1} → E_{−1,t}^{1} is surjective in a range. First, for t ⩽ k − 1 note that the differentials hitting E_{−1,t}^{□} have source E_{j−1,t−j+1}^{j} for 1 ⩽ j ⩽ t + 1. By (A.2) these groups are all zero, so E_{−1,t}^{1} = E_{−1,t}^{∞}. The spectral sequence E converges to zero in total degree t − 1 so E_{−1,t}^{1} = E_{−1,t}^{∞} = 0 and so the first differential d_{1} : E_{0,t}^{1} → E_{−1,t}^{1} is vacuously surjective. For t = k we use the result of the diagram chase above, which tells us that the only possible non-zero differential hitting E_{−1,k}^{□} is the first differential. We know that E_{−1,k}^{∞} = 0 since E converges to zero in total degree k − 1 so the first differential d_{1} : E_{0,k}^{1} → E_{−1,k}^{1} must be surjective. This can be identified as the map on homology induced by the augmentation map R_{k}(M)^{0} → R_{k}(M). Hence we have established:

Fact A.3 The augmentation map a: R_{k}(M)^{0} → R_{k}(M) induces surjections on A-homology up to degree k.

Now consider the maps p: C_{k}(M) → C_{k}(M_{1}) and u: C_{k}(M_{1}) → C_{k}(M), defined as follows. The map p is defined similarly to the stabilisation map. Write M = int(\overline{M}) for a manifold \overline{M} with non-empty boundary and choose a selfembedding e′: \overline{M} ↪ \overline{M} which is isotopic to the identity and whose image does not contain the missing point of M_{1}. Then p is defined by applying e′ to each point of the configuration. The map u is simply the map induced by the inclusion M_{1} ↪ M. Since e′ is isotopic to the identity the composition u ∘ p is homotopic to the identity, and so the induced maps u_{∗} and p_{∗} on homology are semi-inverses: u_{∗} ∘ p_{∗} = id. If we are careful to define p using a self-embedding e′: \overline{M} ↪ \overline{M} whose support is disjoint from the self-embedding e: \overline{M} ↪ \overline{M} used to define the stabilisation map s, then p commutes on the nose with s and there are induced maps p: R_{k}(M) → R_{k}(M_{1}) and u: R_{k}(M_{1}) → R_{k}(M) on mapping cones. Again we have u ∘ p ≃ id so u_{∗} ∘ p_{∗} = id. The methods of the proof of Proposition 6.3 in [Randal-Williams, 2013a] show that

hconn_{A}(u: R_{k−1}(M_{1}) → R_{k−1}(M)) ⩾ ⩾ hconn(s: C_{k−2}(M) → C_{k−1}(M)) + dim(M)

where hconn_{A}(f) is the A-homology-connectivity of f, i.e. the largest ∗ such that H̃_{∗}(mc(f); A) = 0, where mc(f) is the mapping cone of f. By inductive hypothesis the right-hand side is at least k − 2 + dim(M) ⩾ k + 1 since we have assumed that M is at least 3-dimensional. Therefore the A-homology-connectivity of p: R_{k−1}(M) → R_{k−1}(M_{1}) is at least k. In particular we have:

Fact A.4 The map p: R_{k−1}(M) → R_{k−1}(M_{1}) induces surjections on Ahomology up to degree k.

For our third and final fact, consider the spectral sequence ^{0}Ẽ with l = k and recall from just before (A.1) that ^{0}Ẽ_{s,t}^{2} = 0 for t ⩽ k − 1. This is the relative Serre spectral sequence for the map of fibre bundles C_{k}(M)^{0} → C_{k+1}(M)^{0} over C̃_{1}(M) = M. The inclusion of the fibre over a point ∗ ∈ M is the map C_{k−1}(M_{1}) = C_{k−1}(M ∖ {∗}) → C_{k}(M)^{0} which adds the point ∗ to a configuration and labels it by 0. This induces a map f: R_{k−1}(M_{1}) → R_{k}(M)^{0} on mapping cones. The map on H̃_{∗} induced by f can be identified with the composition of the edge homomorphism

H̃_{∗}(R_{k−1}(M_{1})) = ^{0}Ẽ_{0,∗}^{2} ↠ ^{0}Ẽ_{0,∗}^{∞}

and the inclusion

0Ẽ_{0,∗}^{∞} ↪ H̃_{∗}(R_{k}(M)^{0})

given by all the extension problems in total degree ∗. But since the second page is trivial for t ⩽ k − 1 there are no extension problems in total degree ∗ ⩽ k, and so this inclusion is an isomorphism. Hence we have:

Fact A.5 The map f: R_{k−1}(M_{1}) → R_{k}(M)^{0} induces surjections on Ahomology up to degree k.

The composition s′ := a ∘ f ∘ p: R_{k−1}(M) → R_{k}(M) is defined exactly like the stabilisation map s: R_{k−1}(M) → R_{k}(M) except that it uses the self-embedding e′ of \overline{M} instead of e. Since we chose e and e′ to have disjoint support, the maps s and s′ commute. If we now ensure that we picked e and e′ to be isotopic, we have that s and s′ are homotopic. The square s ∘ s′ = s′ ∘ s induces a map of long exact sequences:

H̃_{t}(C_{k}(M)) H̃_{t}(R_{k−1}(M)) H̃_{t−1}(C_{k−1}(M)) H̃_{t−1}(C_{k}(M)) H̃_{t}(C_{k+1}(M)) H̃_{t}(R_{k}(M)) cb = s_{∗} d s′_{∗} a = s′_{∗} s_{∗} s_{∗}

Let t ⩽ k – our aim is to show that H̃_{t}(R_{k}(M)) = 0. By Facts A.3, A.4 and A.5 above, the map a in this diagram is surjective. As mentioned at the beginning of the proof, the stabilisation map is split-injective on homology in all degrees [McDuff, 1975, page 103], so the map b is injective, and so by exactness the map c is surjective. Hence the composite a ∘ c is surjective. But

a ∘ c = d ∘ s′_{∗} = d ∘ s_{∗} = 0,

so its codomain H̃_{t}(R_{k}(M)) must be trivial. It now remains to prove the claim we made earlier in the proof, namely that the map

jĒ_{j,k−j}^{1} = H̃_{k−j}(R_{k−j−1}(M) ∧ R_{j}(ℝ^{n})^{j}) ⟶ H̃_{k−j}(R_{k}(M)^{j}) = E_{j,k−j}^{1}

is surjective. In fact we will show that the map R_{k−j−1}(M) ∧ R_{j}(ℝ^{n})^{j} → R_{k}(M)^{j} induces surjections on homology in degrees t ⩽ k − j. First note that

R_{j}(ℝ^{n})^{j} = mc(C_{j}(ℝ^{n})^{j} → C_{j+1}(ℝ^{n})^{j})
= mc(∅ → C̃_{j+1}(ℝ^{n}))
= C̃_{j+1}(ℝ^{n})_{+}

and the map R_{k−j−1}(M) ∧ C̃_{j+1}(ℝ^{n})_{+} → R_{k}(M)^{j} is given by taking mapping cones of the horizontal arrows in the commutative square:

C_{k−j−1}(M) × C̃_{j+1}(ℝ^{n}) C_{k−j}(M) × C̃_{j+1}(ℝ^{n}) C_{k}(M)^{j} C_{k+1}(M)^{j} s × id s

To do this we begin by defining some more explicit models for various maps. As before, write M = int(\overline{M}) for a manifold \overline{M} with non-empty boundary and choose two isotopic self-embeddings e, e′: \overline{M} ↪ \overline{M} which are both non-surjective and have disjoint support. Choose an embedding ϕ: ℝ^{n} ↪ M ∖ e′(\overline{M}) and pairwise disjoint points p_{0}, …, p_{j} ∈ ℝ^{n}. Write M_{j+1} = M ∖ {ϕ(p_{0}), …, ϕ(p_{j})}. We have a square of maps

(A.4)  _{C_{k−j−1}(M) C_{k−j−1}(M) × C̃_{j+1}(ℝ^{n}) C_{k−j−1}(M_{j+1}) C_{k}(M)^{j} α β γ δ}

defined by

α(c) = (c, (p_{0}, …, p_{j}))
γ(c) = e′(c)
β(c) = c ∪ {ϕ(p_{0}), …, ϕ(p_{j})}; i ↦ ϕ(p_{i})
δ(c, (q_{0}, …, q_{j})) = e′(c) ∪ {ϕ(q_{0}), …, ϕ(q_{j})}; i ↦ ϕ(q_{i}).

Choose a point ∗ ∈ M ∖ e(\overline{M}) and take an explicit model for the stabilisation map to be defined by c ↦ e(c) ∪ {∗}. Since e and e′ have disjoint support this induces a map of squares from (A.4) to (A.4)[k ↦ k + 1]. Taking mapping cones along this map of squares gives us the following:

R_{k−j−1}(M) R_{k−j−1}(M) ∧ C̃_{j+1}(ℝ^{n})_{+} R_{k−j−1}(M_{j+1}) R_{k}(M)^{j} ᾱ β̄ γ̄ δ̄

We need to show that δ̄ induces surjections on homology in degrees t ⩽ k − j. This will follow if we can prove this for γ̄ and β̄. But γ̄ is the composition of j + 1 instances of the map p from Fact A.4, and so this does induce surjections on homology up to degree k − j by Fact A.4.^{3} When j = 0 the map β̄ is surjective on homology up to degree k by Fact A.5. Moreover, the argument proving Fact A.5 generalises (using the spectral sequence ^{j}Ẽ instead of ^{0}Ẽ) to prove precisely that the map β̄ is surjective on homology up to degree k − j in general. □

3The proofs of Facts A.4 and A.5 earlier did not depend on the claim which we are currently proving, so this is not circular.

Remark A.6 When dim(M) ⩾ 3 we have homological stability in the range ∗ ⩽ k for ℚ coefficients (by [Randal-Williams, 2013a, Theorem B]) and for ℤ/p coefficients with p odd (by Proposition A.2 above). Using the short exact sequences of coefficients 0 → ℤ/p → ℤ/p^{l+1} → ℤ/p^{l} → 0 and

0 → ℤ[\frac{1}{2}] → ℚ → ⨁_{p≠2} colim_{l→∞} ℤ/p^{l} → 0

this implies homological stability in the range ∗ ⩽ k − 1 for ℤ[\frac{1}{2}] coefficients. This recovers Theorem 1.4 of [Kupers and Miller, 2014b], except without surjectivity in degree k.

Appendix B. Homological stability for configuration spaces with labels in a fibre bundle

Definition B.1 (Configuration spaces and stabilisation maps with labels in a fibre bundle) Let θ: E → M be a fibre bundle with path-connected fibres F and define

C_{k}(M; θ) := {{p_{1}, …, p_{k}} ⊂ E | θ(p_{i}) ≠ θ(p_{j}) for i ≠ j}.

Choose a self-embedding e: \overline{M} ↪ \overline{M} which is non-surjective and isotopic to the identity. Choose an open neighbourhood U ⊆ \overline{M} containing the support of e, write V = U ∖ ∂\overline{M} and choose a trivialisation ϕ: θ^{−1}(V) → V × F of E over V. Define a self-embedding ẽ: E ↪ E by

p ↦ \begin{cases}p & p ∉ θ^{−1}(V) \\ ϕ^{−1} ∘ (e × id) ∘ ϕ(p) & p ∈ θ^{−1}(V) \end{cases}

and note that θ ∘ ẽ = e ∘ θ. Also choose points ∗ ∈ M ∖ e(\overline{M}) ⊆ V and x ∈ F. We can then define the stabilisation map C_{k}(M; θ) → C_{k+1}(M; θ) by

{p_{1}, …, p_{k}} ↦ {ẽ(p_{1}), …, ẽ(p_{k}), ϕ^{−1}(∗, x)}.

We may generalise Proposition A.2 to configuration spaces with labels in θ using the following fact.

Remark B.2 Configuration spaces also satisfy homological stability with respect to finite-degree twisted coefficient systems: for the case of symmetric groups this was proved by [Betley, 2002], and the general case was proved in [Palmer, 2013]. A twisted coefficient system for M is a functor from the partial braid category B(M) to ℤ-mod. The partial braid category B(M) has objects {0, 1, 2, … } and a morphism from m to n is a path in C_{k}(M) from a subset of {p_{1}, …, p_{m}} to a subset of {p_{1}, …, p_{n}}, up to endpoint-preserving homotopy, where {p_{1}, p_{2}, p_{3}, … } is a fixed injective sequence in M. If the twisted coefficient system has degree d the stable range obtained is ∗ ⩽ \frac{k−d}{2}, which arises since homological stability with untwisted ℤ coefficients in the range ∗ ⩽ \frac{k}{2} is an input for the proof. However, if the twisted coefficient system takes values in the subcategory ℤ[\frac{1}{2}]-mod of ℤ-mod and dim(M) ⩾ 3, then we may instead input [Kupers and Miller, 2014b] or Proposition A.2 to obtain a stable range of ∗ ⩽ k − d for C_{k}(M) with coefficients in a functor B(M) → ℤ[\frac{1}{2}]-mod of degree d.

Proposition B.3 Let M be a smooth, connected, open n-manifold with n ⩾ 2 and θ: E → M a fibre bundle with path-connected fibres. Then the stabilisation map C_{k}(M; θ) → C_{k+1}(M; θ) induces isomorphisms on H_{∗}(−; ℤ) in the range ∗ ⩽ \frac{k}{2} − 1. It induces isomorphisms in the range ∗ ⩽ k on H_{∗}(−; ℚ), unless M is an orientable surface in which case the range is only ∗ ⩽ k − 1. If n ⩾ 3 it induces isomorphisms on H_{∗}(−; ℤ[\frac{1}{2}]) in the range ∗ ⩽ k − 1.

The worse range ∗ ⩽ k−1 for rational homology of configurations on orientable surfaces is necessary: for example H_{1}(C_{1}(ℝ^{2}); ℚ) = 0 ≅ ℚ̸ ≅ H_{1}(C_{2}(ℝ^{2}); ℚ).

Remark B.4 A version of Proposition B.3 is also proved in the appendix of [Kupers and Miller, 2014a]. One of the proofs given there is essentially the same as the proof we give below, and a sketch proof using semi-simplicial resolutions by collections of disjoint arcs in M is also given. This latter method has the advantage that it gives a range of ∗ ⩽ \frac{k}{2} for ℤ coefficients (at least when M is orientable), rather than the smaller range ∗ ⩽ \frac{k}{2} − 1 for ℤ coefficients obtained in Proposition B.3.

Proof. This will follow by the same considerations as in Remark A.6 if we show that it induces isomorphisms on H_{∗}(−; A) in the range ∗ ⩽ \frac{k}{2} when A = 𝔽_{p}, in the range ∗ ⩽ k if either (a) A = ℚ and M is not an orientable surface or (b) A = 𝔽_{p} for p odd and n ⩾ 3, and in the range ∗ ⩽ k − 1 when A = ℚ and M is an orientable surface. The loss of one degree from the range occurs when going from ℚ and ℚ/ℤ coefficients to ℤ coefficients (resp. ℚ and ℚ/ℤ[\frac{1}{2}] coefficients to ℤ[\frac{1}{2}] coefficients). Let A = ℚ or 𝔽_{p} for a prime p. There are fibre bundles C_{k}(M; θ) → C_{k}(M), given by forgetting labels, with fibre F^{k}. The stabilisation maps C_{k}(M) → C_{k+1}(M) and C_{k}(M; θ) → C_{k+1}(M; θ) commute with these fibre bundles and the map of fibres is the inclusion F^{k} ↪ F^{k+1}. There is then a map of Serre spectral sequences

E_{s,t}^{2} ≅ H_{s}(C_{k}(M); H_{t}(F^{k}; A)) H_{∗}(C_{k}(M; θ); A) E_{s,t}^{2} ≅ H_{s}(C_{k+1}(M); H_{t}(F^{k+1}; A)) H_{∗}(C_{k+1}(M; θ); A) ⇒ ⇒

and our aim is to prove that the map in the limit is an isomorphism in a certain range depending on A and M. Now by Lemma 4.2 of [Palmer, 2013] and since A is a field the assignment k ↦ H_{t}(F^{k}; A) extends to form a twisted coefficient system of degree at most \frac{t}{h+1} ⩽ t where h = hconn_{A}(F). Hence the map on the second page is an isomorphism in the range s ⩽ \frac{k−t}{2} by Theorem 1.3 of [Palmer, 2013]. In particular it is an isomorphism in total degree at most \frac{k}{2} and therefore the same holds for the map in the limit. To obtain the improved range in certain cases note that, by Remark 6.5 of [Palmer, 2013], if homological stability with (untwisted) A coefficients holds for (unlabelled) configuration spaces on M in the range ∗ ⩽ f(k), then twisted homological stability will hold in the range ∗ ⩽ f(k−d) for any twisted coefficient system of degree d which factors through the forgetful functor A-mod → ℤ-mod (c.f. Remark B.2). If A = ℚ then the above twisted coefficient system factors through the inclusion ℚ-mod → ℤ-mod. By Theorem C of [Randal-Williams, 2013a] we may take f(k) = k if n = dim(M) ⩾ 3. For orientable surfaces we may take f(k) = k − 1 by Corollary 3 of [Church, 2012] or Theorem 1.3 of [Knudsen, 2014], and for non-orientable surfaces we may take f(k) = k by Theorem 1.3 of [Knudsen, 2014]. By the above paragraph the map of spectral sequences is an isomorphism on the second page in the range s ⩽ f(k − t), and therefore in total degree at most k (resp. total degree at most k − 1 for orientable surfaces). Hence so is the map in the limit. If A = 𝔽_{p} for p odd and n ⩾ 3 then the twisted coefficient system factors through the inclusion ℤ[\frac{1}{2}]-mod → ℤ-mod. By Proposition A.2 (or Theorem 1.4 of [Kupers and Miller, 2014b]) we may take f(k) = k. So as above the map of spectral sequences is an isomorphism in total degree at most k, and therefore so is the map in the limit. □

Appendix C. Stable homology of configuration spaces with labels in a fibre bundle

In this appendix we prove Theorem 4.6. Another proof can be obtained adapting step by step the proof in [McDuff, 1975] for trivial labels, as pointed out in the introduction to that paper. We give here a sketch of this proof with some shortcuts, taking advantage of knowing the homology stability theorem with labels (Proposition B.3) in the spirit of [Galatius et al., 2009].

Definition C.1 Let M be an open manifold, let c: D^{n−1} × (0, 1] be a proper embedding and let M_{1} = M ∪_{c} (D̊^{n−1} × (−1, 1]). Define ψ^{δ,γ}(M; θ) to be the space whose underlying set is C^{δ,γ}(M; θ) := ∐_{k} C_{k}^{δ,γ}(M; θ) with the following topology: Consider the quotient Y of C^{δ,γ}(M_{1}; θ) under the relation ∼ where (q, ϵ, {f_{q}}_{q∈q}) ∼ (q′, ϵ′, {f′_{q′}}_{q′∈q′}) if and only if

(1) q ∩ M = q′ ∩ M,

(2) if the above intersection is non-empty, then ϵ = ϵ′,

(3) if the above intersection is non-empty, then f_{q} = f′_{q} for all q ∈ q ∩ M.

The natural inclusion C^{δ,γ}(M; θ) → C^{δ,γ}(M_{1}; θ) induces a bijection C^{δ,γ}(M; θ) ≅ Y, which we use to endow C^{δ,γ}(M; θ) with a new topology.

Recall that we defined the non-linear scanning map with labels

𝔰^{θ,γ}: C_{k}^{δ,γ}(M; θ) ⟶ Γ_{c}(ψ^{δ}(T^{1}M; θ))

in §4. Following the same recipe we can define a scanning map

𝔰^{θ,γ}: ψ^{δ,γ}(M; θ) ⟶ Γ(ψ^{δ}(T^{1}M; θ))

whose target is now the whole space of sections.

Lemma C.2 ([Hesselholt, 1992, §2.3]) If M is connected, then the scanning map is a homotopy equivalence.

In the paper, Hesselholt considers E to be a fibre bundle of based spaces. This lemma is a particular case of his theorem, taking a disjoint basepoint in each fibre and the submanifold N in his theorem to be connected. Let π_{1} : M −−→ I be the partially defined function that sends a point in the image of D^{n−1} × I to the second coordinate. Define ψ^{δ,γ}(M; θ)_{•} to be the semi-simplicial space whose space of i-simplices is the space of tuples (q, ϵ, {f_{q}}_{q∈q}, a_{0}, …, a_{i}), where (q, ϵ, {f_{q}}_{q∈q}) ∈ ψ^{δ,γ}(M; θ) and (a_{0}, …, a_{i}) ∈ I^{i+1} and π_{1}(q) ∩ {a_{0}, …, a_{i}} = ∅. The jth face map forgets a_{j}, and there is an augmentation to ψ^{δ,γ}(M; θ) that forgets all the a_{j}’s.

Lemma C.3 The realization of the augmentation

∥ψ^{δ,γ}(M; θ)_{•}∥ → ψ^{δ,γ}(M; θ)

is a weak homotoy equivalence.

Proof. This is an augmented topological flag complex [Galatius and Randal-Williams, 2014] satisfying the conditions of Theorem 6.2 in that paper, hence a weak homotopy equivalence. □

Proposition C.4 If θ: E → M has path-connected fibres, then the restriction of the scanning map

𝔰^{γ,θ}: C_{k}^{δ,γ}(M ∖ c; θ) ⟶ Γ_{c}(Ψ^{δ}(T^{1}M ∖ c; θ))

is a homology isomorphism in the range in which the stabilisation map of Proposition B.3 is a homology isomorphism. Since M ∖ c ≅ M, the same holds for M.

Proof. We have constructed the following commutative diagram:

\xymatrix{∥ψ^{δ,γ}(M; θ)_{•}∥ \ar[r] \ar[d] & ψ^{δ,γ}(M; θ) \ar[r] \ar[d] & Γ(Ψ^{δ}(T^{1}M; θ)) \ar[d] \\ ∥ψ^{δ,γ}(D^{n−1} × I; θ)_{•}∥ \ar[r] & ψ^{δ,γ}(D^{n−1} × I; θ) \ar[r] & Γ(ψ^{δ}(T^{1}(D^{n−1} × I); θ))} (C.1)

All the horizontal maps are homotopy equivalences, by the previous two lemmas. The rightmost vertical map is a fibration. We now choose another properly embedded ray D^{n−1} × (0, 1] ≅ L ⊂ ∂M ∖ c, and take the colimit

P (M; θ)_{•} := colim (ψ^{δ,γ}(M; θ)_{•} ⟶^{s} ψ^{δ,γ}(M; θ)_{•} ⟶ …)

with respect to the stabilisation maps s that push the configurations outside L and adds a point in L with some prescribed label. The scanning of this operation 𝔰^{γ,θ}(s) gives also a sequence of maps between spaces of sections, whose colimit we denote by

G(M; θ) := colim (Γ(ψ^{δ}(T^{1}M; θ)) ⟶^{𝔰^{θ,γ}(s)} Γ(ψ^{δ}(T^{1}M; θ)) ⟶ …)

Observe that the maps 𝔰^{γ,θ}(s) increase the degree by 1. We can consider instead the maps that push the source of the scanning map away from L and glue there the reflection of the scanning of some point in L, together with some prescribed label. This latter map is a homotopy inverse of 𝔰^{γ,θ}(s), hence the maps 𝔰^{γ,θ}(s) are homotopy equivalences. By Proposition B.3, it follows that the semi-simplicial map

P (M; θ)_{•} ⟶ ψ^{δ,γ}(D^{n−1} × I; θ)_{•}

satisfies the hypotheses of [McDuff and Segal, 7576, Proposition 4], so the realization

∥P (M; θ)_{•}∥ ⟶ ∥ψ^{δ,γ}(D^{n−1} × I; θ)_{•}∥

is a homology fibration. Its fibre over any point is the colimit of the space C^{δ,γ}(M; θ) with respect to the stabilisation map s. The map

G(M; θ) ⟶ Γ(ψ^{δ,γ}(T^{1}M; θ))

is a Serre fibration (it is a union of Serre fibrations). Its fibre over any point is the colimit of Γ_{c}(ψ^{δ}(T^{1}M); θ) with respect to the map obtained by scanning s:

(C.2)  \xymatrix{colim C^{δ,γ}(M; θ) \ar[r] \ar[d] & colim Γ_{c}(ψ^{δ}(T^{1}M); θ) \ar[d] \\ ∥P (M; θ)_{•}∥ \ar[r]^{≃} \ar[d]^{a} & G(M; θ) \ar[d]^{b} \\ ∥ψ^{δ,γ}(D^{n−1} × I; θ)_{•}∥ \ar[r]^{≃} & Γ(ψ^{δ}(T^{1}M; θ))}

The fibres of (C.2) together with the maps to the fibres of (C.2) give the following commutative diagram

\xymatrix{C^{δ,γ}(M; θ) \ar[r]^{𝔰^{γ,θ}} \ar[d] & Γ_{c}(ψ^{δ}(T^{1}M); θ) \ar[d] \\ colim C^{δ,γ}(M; θ) \ar[r] & colim Γ_{c}(ψ^{δ}(T^{1}M); θ).}

Documenta Mathematica 20 (2015) 753–805

The bottom map is a homology equivalence because the horizontal maps in diagram (C.1) are homotopy equivalences. The left vertical map is a homology equivalence in the stable range of Proposition B.3. The right vertical map is a homotopy equivalence. As a consequence, the upper horizontal map is a homology equivalence in the range provided by Proposition B.3 □

The following is proved in the same way as Theorem 1.1 at the bottom of page 34 in [McDuff, 1975].

Corollary C.5 If M is a manifold with empty boundary, then the non-linear scanning map

𝔰^{θ,γ}: C^{δ,γ}(M; θ) ⟶ Γ_{c}(ψ^{δ}(T^{1}M; θ))

is a homology isomorphism in the range in which the stabilisation map is a homology isomorphism.

We showed in Section §4 that the triangle

(C.3)  _{C_{k}^{δ,γ}(M; θ) Γ_{c}(ψ^{δ}(T^{1}M; θ)) Γ_{c}(Ṫ^{1,θ}M). 𝔰^{θ,γ} 𝒮 h i≃}

commutes, hence from Corollary C.5 it follows that:

Theorem C.6 (McDuff’s Theorem with labels) The linear scanning map with labels

𝒮^{δ,θ}: C_{k}(M; θ) ⟶ Γ_{c}(Ṫ^{θ}(M))_{k}

induces an isomorphism on homology groups in the stable range provided by Proposition B.3.

References

[Adams, 1960] Adams, J. F. (1960). On the non-existence of elements of Hopf invariant one. Ann. of Math. (2), 72:20–104. [Bendersky and Miller, 2014] Bendersky, M. and Miller, J. (2014). Localization and homological stability of configuration spaces. Q. J. Math., 65(3):807–815. [Berrick et al., 2006] Berrick, A. J., Cohen, F. R., Wong, Y. L., and Wu, J. (2006). Configurations, braids, and homotopy groups. J. Amer. Math. Soc., 19(2):265–326. [Betley, 2002] Betley, S. (2002). Twisted homology of symmetric groups. Proc. Amer. Math. Soc., 130(12):3439–3445 (electronic). [Blanchet and Marin, 2007] Blanchet, C. and Marin, I. (2007). Cabling Burau representation. arXiv:math/0701189v1. [Bödigheimer et al., 1989] Bödigheimer, C.-F., Cohen, F., and Taylor, L. (1989). On the homology of configuration spaces. Topology, 28(1):111–123. [Church, 2012] Church, T. (2012). Homological stability for configuration spaces of manifolds. Invent. Math., 188(2):465–504.

[Cohen et al., 1976] Cohen, F. R., Lada, T. J., and May, J. P. (1976). The homology of iterated loop spaces. Springer-Verlag, Berlin. Lecture Notes in Mathematics, Vol. 533. [Dold, 1963] Dold, A. (1963). Partitions of unity in the theory of fibrations. Ann. of Math. (2), 78:223–255. [Fadell and Van Buskirk, 1962] Fadell, E. and Van Buskirk, J. (1962). The braid groups of E^{2} and S^{2}. Duke Math. J., 29:243–257. [Félix and Thomas, 2000] Félix, Y. and Thomas, J.-C. (2000). Rational Betti numbers of configuration spaces. Topology Appl., 102(2):139–149. [Galatius et al., 2009] Galatius, S., Madsen, I., Tillmann, U., and Weiss, M. (2009). The homotopy type of the cobordism category. Acta Math., 202(2):195–239. [Galatius and Randal-Williams, 2014] Galatius, S. and Randal-Williams, O. (2014). Stable moduli spaces of high-dimensional manifolds. Acta Math., 212(2):257–377. [Hansen, 1974] Hansen, V. L. (1974). The homotopy problem for the components in the space of maps on the n-sphere. Quart. J. Math. Oxford Ser. (2), 25:313–321. [Hansen, 1981] Hansen, V. L. (1981). On spaces of maps of n-manifolds into the n-sphere. Trans. Amer. Math. Soc., 265(1):273–281. [Hesselholt, 1992] Hesselholt, L. (1992). A homotopy theoretical derivation of Q Map (K, −)_{+}. Math. Scand., 70(2):193–203. [Knudsen, 2014] Knudsen, B. (2014). Betti numbers and stability for configuration spaces via factorization homology. arXiv:1405.6696v4. [Kupers and Miller, 2014a] Kupers, A. and Miller, J. (2014a). E_{n}-cell attachments and a local-to-global principle for homological stability. arXiv:1405.7087v2. [Kupers and Miller, 2014b] Kupers, A. and Miller, J. (2014b). Improved homological stability for configuration spaces after inverting 2. arXiv:1405.4441v3. [Martin and Woodcock, 2003] Martin, P. P. and Woodcock, D. (2003). Generalized blob algebras and alcove geometry. LMS J. Comput. Math., 6:249–296. [McDuff, 1975] McDuff, D. (1975). Configuration spaces of positive and negative particles. Topology, 14:91–107. [McDuff and Segal, 7576] McDuff, D. and Segal, G. (1975/76). Homology fibrations and the “group-completion” theorem. Invent. Math., 31(3):279–284. [Milgram and Löffler, 1988] Milgram, R. J. and Löffler, P. (1988). The structure of deleted symmetric products. In Braids (Santa Cruz, CA, 1986), volume 78 of Contemp. Math., pages 415–424. Amer. Math. Soc., Providence, RI. [Møller, 1987] Møller, J. M. (1987). Nilpotent spaces of sections. Trans. Amer. Math. Soc., 303(2):733–741. [Nagpal, 2015] Nagpal, R. (2015). FI-modules and the cohomology of modular representations of symmetric groups. arXiv:1505.04294v1. [Palmer, 2013] Palmer, M. (2013). Twisted homological stability for configuration spaces. arXiv:1308.4397v2.

[Randal-Williams, 2013a] Randal-Williams, O. (2013a). Homological stability for unordered configuration spaces. Q. J. Math., 64(1):303–326. [Randal-Williams, 2013b] Randal-Williams, O. (2013b). “Topological chiral homology” and configuration spaces of spheres. Morfismos, 17(2):57–70. [Salvatore, 2004] Salvatore, P. (2004). Configuration spaces on the sphere and higher loop spaces. Math. Z., 248(3):527–540. [Segal, 1979] Segal, G. (1979). The topology of spaces of rational functions. Acta Math., 143(1-2):39–72. [Serre, 1951] Serre, J.-P. (1951). Homologie singulière des espaces fibrés. Applications. Ann. of Math. (2), 54:425–505. [Whitehead, 1946] Whitehead, G. W. (1946). On products in homotopy groups. Ann. of Math (2), 47:460–475. [Whitehead, 1953] Whitehead, J. H. C. (1953). On certain theorems of G. W. Whitehead. Ann. of Math. (2), 58:418–428.

Federico Cantero

Mathematisches Institut Universität Münster Einsteinstraße 62 48149 Münster Germany fcant 01@uni-muenster.de Martin Palmer Mathematisches Institut Universität Münster Einsteinstraße 62 48149 Münster Germany mpalm 01@uni-muenster.de 806

Documenta Mathematica 20 (2015)