Homology, Homotopy and Applications, vol.6(1), 2004, pp.153–166

HOMOLOGY FIBRATIONS AND ”GROUP-COMPLETION” REVISITED

WOLFGANG PITSCH and JÉRÔME SCHERER

(communicated by Dominique Arlettaz)

Abstract We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but follows a modern treatment of homotopy colimits, using as little simplicial technology as possible. We compare simplicial and topological definitions of homology fibrations.

Introduction

The group completion of a topological monoid M is the loop space ΩBM and a group completion theorem is originally a statement about the relation between the homology of M and that of ΩBM. In the appendix of [FM94] D. Quillen considers a simplicial monoid M. His main theorem is that under certain conditions the homology of the group completion of M can be computed by inverting π_{0}M in the homology of M. A similar result can be found in May’s [May75, Theorem 15.1]. In this paper we focus on a more topological kind of group completion theorem, the question being how to construct ΩBM out of M. Our starting point is McDuff’s and Segal’s theorem, as it can be found in [MS76, Proposition 2] (a good account on the subject is Adams’ book on infinite loop spaces [Ada78, Chapter 3]).

Theorem. Let M be a topological monoid acting on a space X by homology equivalences. Then the map π : EM ×_{M} X → BM from the Borel construction to the classifying space of M is a homology fibration with fiber X.

The standard application is as follows. Let M be a homotopy commutative topological monoid with π_{0}M ≅ ℕ. Choose a point m in the component of 1 and form the telescope M_{∞} = T el(M ⟶^{·m} M ⟶^{·m} …). The action of M by left multiplication on M_{∞} is by homology equivalences because M is homotopy commutative. Hence one obtains: Corollary. Let M be a homotopy commutative topological monoid. Then there is a homology equivalence M_{∞} → ΩBM. Moreover, when π_{1}M_{∞} is perfect, ΩBM ≃ M_{∞}^{+}.

The first author was partially supported by the FNRS grant 2000-059008.99/1 and the second by the program Ramón y Cajal, MCyT, Spain. Received July 25, 2003, revised March 2, 2004; published on April 21, 2004. 2000 Mathematics Subject Classification: Primary 55U10; Secondary 19D06 Key words and phrases: Group completion, homology fibration, plus-construction, barycentric subdivision, simplex category. \begin{matrix}c \\ ◯ \end{matrix} 2004, Wolfgang Pitsch and Jérôme Scherer. Permission to copy for private use granted.

We discuss in Theorem 3.5 a strengthened version of the corollary which takes into account cases when the fundamental group is not perfect. Taking for example M to be the disjoint union ∐ BΣ_{n} of classifying spaces of the symmetric groups, the Barrat-Priddy-Quillen Theorem states that BΣ_{∞}^{+} is the infinite loop space QS^{0}, [BP72]. Likewise, taking M to be ∐ BGL_{n}(R) one gets back Quillen’s definition of the algebraic K-theory of a ring R, [Qui71].

Simplicial versions of the group completion theorem started appearing at the end of the eighties. I. Moerdijk provides a homological statement in [Moe89, Corollary 3.1] and J.F. Jardine the analogue of the above theorem in [Jar89, Theorem 4.2], which he calls the “strong form of the Group Completion Theorem”. More recently U. Tillmann introduced a “multiple object case” in her celebrated work on the stable mapping class group ([Til97, Theorem 3.2]). In this context the Borel construction is replaced by a bisimplicial version, i.e. the realization of a certain simplicial space. Let M be a simplicial category and F : M^{op} → Spaces a contravariant diagram. There is always a natural transformation to the trivial diagram. Taking the bisimplicial Borel constructions yields a map π_{M} : E_{M}F → BM, analogous to the map π in the classical theorem.

Theorem 3.2. Let M be a simplicial category and F : M^{op} → Spaces a contravariant diagram. Assume that any morphism f : i → j induces an isomorphism in integral homology H_{∗}(F (j); ℤ) → H_{∗}(F (i); ℤ). Then, for each object i ∈ M, the map F (i) → F ib_{i}(π_{M}) to the homotopy fiber of π_{M} over i is a homology equivalence.

We offer in this paper a direct proof which uses as little simplicial technology as possible. In comparison, Jardine’s proof makes use of bisimplex categories, trisimplicial maps, and a homology spectral sequence (the Bousfield-Kan spectral sequence is required if one wishes to prove the analogue theorem for an arbitrary homology theory) and Tillmann’s theorem builds up on this result. Our main ingredient is a rather classical result about comparing the fiber of the realization with the realization of the fibers, an idea already used by McDuff and Segal in their proof of the classical group completion theorem. Of course we do not avoid simplicial spaces, the theorem after all is about delooping a simplicial classifying space. We work however more in the spirit of the modern theory homotopy colimits. One very powerful tool in this setting is to decompose a space as a diagram over its simplices. The advantage of this approach is that one gets a more geometric feeling about the constructions performed (such as the bisimplicial Borel construction). In particular the generalization proposed in Remark 3.6 in the context of an arbitrary homology theory is straightforward.

We use in this paper a simplicial notion of homology fibrations : preimages of simplices have the same homology as the homotopy fiber. The word space means simplicial set and we write Spaces for the category of spaces. In the last section we compare this concept to that of classical homology fibration in the category of topological spaces and prove they coincide.

Acknowledgements: We would like to thank J.B. Bost, F. Loeser, and I. Madsen for organizing the week about the mapping class group in Luminy (January 2002). This work originated at the university of Lausanne while we were preparing our talks. We are indebted to W. Chachólski for the helpful comments he made on an early version of the manuscript, as well as for introducing the second author to simplicial technology.

1. Homology fibrations

We begin by recalling some basic definitions. The standard n-simplex is denoted by Δ[n]. It has a unique non-degenerate simplex in dimension n. A map of spaces Δ[n] → B is determined by the image of this simplex, which is an n-simplex of B. The simplex category of a space B is the category ΔB whose objects are all simplices σ : Δ[n] → B of B and the morphisms are commutative triangles (this category is defined in [Seg74, p. 308], see also [DF96, p.182], and [CS02, Definition 6.1]). In particular there is always a morphism d_{i}σ → σ, where d_{i}σ is the i-th face of the simplex σ. This allows to decompose any map p : E → B as a diagram over ΔB in the following way. Let σ be an n-simplex in B and denote by dp(σ) the pull-back of the diagram Δ[n] ⟶^{σ} B ⟵^{p} E. This is the “preimage” of the simplex in E and yields a functor dp : ΔB → Spaces. The map p can then be recovered up to homotopy by taking the homotopy colimit over ΔB of the natural transformation dp → ∗, as E ≃ hocolim_{ΔB}dp and B ≃ hocolim_{ΔB}Δ[n]. This last homotopy colimit is equivalent to the nerve of ΔB and G. Segal introduced the simplex category in [Seg74] precisely to explain the geometric realization.

We will also need a slight generalization of dp, replacing a simplex by any space K. For a map f : K → B, define dp(f) to be the pull-back of f along p. By a homology equivalence we mean a map which induces an isomorphism in ordinary homology with integral coefficients.

Definition 1.1. A map of spaces p : E → B is a homology fibration if the natural map dp(σ) → F ib_{σ}(p) to the homotopy fiber of p over the component of σ is a homology equivalence for any simplex σ ∈ B. It is a weak homology fibration if for any simplex σ ∈ B and any simplicial operation θ we have a homology equivalence dp(θσ) → dp(σ).

The aim of this section is to prove that a weak homology fibration is actually a homology fibration. This part of the paper forms a simplicial analogue of McDuff and Segal’s work on locally contractible paracompact spaces.

Lemma 1.2. [MS76, Proposition 6] Let p : E → B be a weak homology fibration with B contractible. Then p is a homology fibration.

Proof. The category ΔB is contractible since B ≃ hocolim_{ΔB}∗ = N(ΔB). So E is equivalent to the homotopy colimit over a contractible category of a diagram in which all maps are homology equivalences. This homotopy colimit has the same homology type as any of the values dp(σ) since it can be computed ([Ami94]) by using only push-outs and telescopes of diagrams consisting of homology equivalences. We conclude by the Mayer–Vietoris Theorem and the fact that homology commutes with telescopes.

Proposition 1.3. Let p : E → B be a weak homology fibration. The pull-back of p along any map f : B′ → B is another weak homology fibration p′ : E′ → B′.

Proof. Let σ′ be a simplex in B′, σ = fσ′ its image in B and θ any simplicial operation. Then dp(σ) has the same homology type as dp(θσ) by assumption. But dp′(σ′) ≃ dp(σ) and dp′(θσ′) ≃ dp(θσ) since p′ was obtained as a pull-back.

Theorem 1.4. [MS76, Proposition 5] A weak homology fibration is a homology fibration.

Proof. Let p : E → B be a weak homology fibration and choose f : P B→ →B the path space fibration. The above proposition applies, so p′ : F ib_{σ}(p) → P B is a weak homology fibration as well for any simplex σ in B. Since f is surjective, there exists a simplex σ′ ∈ P B such that f(σ′) = σ. Therefore dp(σ) ≃ dp′(σ′), which has the same homology type as the homotopy fiber F ib_{σ}(p) by Lemma 1.2.

2. Realizations and fibers

Theorem 1.4 will be used throughout this section. For checking that a map is a homology fibration it suffices to check it is a weak homology fibration.

Lemma 2.1. Consider a commutative square

\xymatrix{E_{0} \ar[r] \ar[d]_{p_{0}} & E_{1} \ar[d]^{p_{1}} \\ B_{0} \ar[r] & B_{1}}

where the vertical arrows are compatible homology fibrations in the sense that the map F ib_{v}(p_{0}) → F ib_{v}(p_{1}) is a homology equivalence for any vertex v ∈ B_{0}. Then dp_{0}(f) → dp_{1}(f) is a homology equivalence for any map f : K → B_{0}. Moreover if both horizontal maps are cofibrations, then so is dp_{0}(f) → dp_{1}(f).

Proof. Notice first that if σ is a simplex in B_{0}, then dp_{0}(σ) → dp_{1}(σ) is a homology equivalence by our assumption on the homotopy fibers over vertices. Likewise the preimages in E_{0} and E_{1} of a disjoint union of simplices have the same integral homology type. We assume therefore that K is connected. Assume K = L ∪_{∂Δ[n]} Δ[n]. By induction on the dimension suppose that both dp_{0}(f|_{L}) → dp_{1}(f|_{L}) and dp_{0}(f|_{∂Δ[n]}) → dp_{1}(f|_{∂Δ[n]}) are homology equivalences. We see that the preimage of ∂Δ[n] is contained in that of Δ[n] so that

dp_{0}(f) = colim(dp_{0}(f|_{L}) ← dp_{0}(f|_{∂Δ[n]}) ↪ dp_{0}(f|_{Δ[n]}))

is actually a homotopy push-out. Thus dp_{0}(f) → dp_{1}(f) is a homotopy push-out of homology equivalences.

We prove now that a push-out of homology fibrations is still a homology fibration. As a map can always be replaced by a fibration, we must pay close attention to the constructions we perform. We always use strict colimits, but for diagrams where the colimit is weakly equivalent to the homotopy colimit (thus in a push-out diagram at least one map will be a cofibration, and in a telescope diagram all maps will be cofibrations).

Proposition 2.2. Consider a natural transformation between push-out diagrams:

[figure: E; p; =; colim (; E_{1}; p_{1}; E_{0}; p_{0}; E_{2}; p_{2}; ); B; =; colim (; B_{1}; B_{0}; B_{2}; )]

such that p_{n} : E_{n} → B_{n} is a homology fibration for 0 ⩽ n ⩽ 2 and the right handside horizontal maps are cofibrations. Assume that the map F ib_{v}(p_{0}) → F ib_{v}(p_{n}) is a homology equivalence for any vertex v ∈ B_{0} if n = 1, 2. Then p is a homology fibration as well. Moreover, if for some 0 ⩽ n ⩽ 2, w is a vertex in B_{n}, then B_{n} ↪ B induces a homology equivalence F ib_{w}(p_{n}) → F ib_{w}(p).

Proof. Any simplex σ in B lies either in B_{1} or in B_{2}. Say it lies in B_{1} (the other case is similar) and consider the pull-back K of Δ[n] → B_{1} ← B_{0}. Apply Lemma 2.1 to the map f : K → B_{0} to conclude that dp_{0}(f) → dp_{2}(f) is a homology equivalence, which is even a cofibration. Hence the preimage dp(σ) is the (homotopy) pushout colim(dp_{1}(σ) ← dp_{0}(f) ↪ dp_{2}(f)). The homotopy push-out of a homology equivalence is again a homology equivalence so that dp(σ) has the same homology type as dp_{1}(σ). We conclude that p is a weak homology fibration.

Proposition 2.3. Consider a natural transformation between telescope diagrams:

[figure: E; f; =; colim (; E_{0}; p_{0}; E_{1}; p_{1}; E_{2}; p_{2}; ⋯; ); B; =; colim (; B_{0}; B_{1}; B_{2}; ⋯; )]

such that p_{n} : E_{n} → B_{n} is a homology fibration for any n ⩾ 0 and all horizontal maps are cofibrations. Assume that the map F ib_{v}(p_{n}) → F ib_{v}(p_{n+1}) is a homology equivalence for any n ⩾ 0 and any vertex v ∈ B_{n}. Then p is a homology fibration as well. Moreover, if w is a vertex in B_{n} for some n ⩾ 0, then the inclusion B_{n} ↪ B induces a homology equivalence F ib_{w}(p_{n}) → F ib_{w}(p).

Proof. As B = ⋃ B_{n}, any simplex σ of B lies in some B_{N}. The conclusion follows since dp(σ) = ⋃_{n⩾N} dp_{n}(σ) has the same homology type as dp_{N}(σ).

Let X_{•} be a simplicial space. Recall that Segal’s thick realization ||X_{•}|| ([Seg74, Appendix A]) is defined by an inductive process. We have ||X_{•}|| = ⋃_{n} ||X_{•}||_{n} where ||X_{•}||_{0} = X_{0} and ||X_{•}||_{n} is constructed from ||X_{•}||_{n−1} by the following push-out

colim(||X_{•}||_{n−1} ← ∂Δ[n] × X_{n} ↪ Δ[n] × X_{n})

and the map ∂Δ[n] × X_{n} → ||X_{•}||_{n−1} is defined using only the face maps. This thick realization can be seen as the homotopy colimit of the diagram X_{•} over the subcategory of Δ^{op} generated by the face morphisms.

Theorem 2.4. [MS76, Proposition 4] Let p_{•} : E_{•} → B_{•} be a map of simplicial spaces such that p_{n} : E_{n} → B_{n} is a weak homology fibration for any n ⩾ 0. Assume that any face map d_{i} : [n] → [n + 1] induces a homology equivalence on homotopy fibers F ib_{v}(p_{n+1}) → F ib_{d_{i}v}(p_{n}) for any vertex v ∈ B_{n+1}. Then p : ||E_{•}|| → ||B_{•}|| is a homology fibration as well. Moreover, if w is a vertex in ||B_{•}|| lying in the same connected component as a vertex v ∈ B_{n}, then there is a homology equivalence F ib_{w}(p_{n}) → F ib_{v}(p).

Proof. Each step is a homotopy push-out involving only the face maps, so Proposition 2.2 applies. Hence ||p_{•}||_{n} is a homology fibration for any n ⩾ 0 and we conclude by Proposition 2.3.

One could actually prove a more general statement involving a colimit over a small indexing category instead of the realization of a simplicial space. In this paper we will not need such a statement.

3. The generalized group completion

The aim is to find a model for the loops on the classifying space of a simplicial category. Let us start with a brief reminder on simplicial categories. More details can be found for example in [Til97, Section 1], especially about the link with 2-categories. Roughly speaking a simplicial category is a category equipped with spaces of morphisms instead of sets of morphisms. So mor_{M}(i, j) is a space for any objects i, j ∈ M and mor_{M}(i, i) contains the identity morphism as distinguished base point. More precisely a simplicial category M is a simplicial object in the category CAT of small categories with constant object set. It is helpful to look at M as a functor Δ^{op} → CAT, where the category of n-simplices is the category having same objects as M and morphisms from i to j are the n-simplices of the space of morphisms from i to j. Taking now the nerve of this simplicial category degree by degree produces a simplicial space denoted by BM_{•}, the simplicial classifying space. classifying space is composable 2-morphisms.

A contravariant diagram F : M^{op} → Spaces is the data of spaces F (i) for all objects i ∈ M and natural continuous maps μ_{i,j} : mor_{M}(i, j) × F (j) → F (i). The simplicial category itself produces an example of a diagram with M(i) = ∐_{j∈Obj(M)} mor_{M}(i, j).

Definition 3.1. The bisimplicial Borel construction of a diagram F : M^{op} → Spaces is the simplicial space E_{M}F_{•} whose space of n-simplices is the disjoint union over all n + 1-tuples of objects in M

∐_{i_{0},...,i_{n}} mor_{M}(i_{n}, i_{n−1}) × ⋯ × mor_{M}(i_{1}, i_{0}) × F (i_{0})

The degeneracy maps are the obvious inclusions. The face map d_{n} : E_{M}F_{n} → E_{M}F_{n−1} is projection on the last n factors, d_{0} = 1 × μ_{i_{1},i_{0}}, and the other d_{k}’s are defined by composition mor_{M}(i_{k+1}, i_{k}) × mor_{M}(i_{k}, i_{k−1}) → mor_{M}(i_{k+1}, i_{k−1}) .

The trivial diagram T (i) = {i} is the diagram in which any morphism i → j induces the unique map {j} → {i}. The bisimplicial Borel construction of the trivial diagram is nothing but the simplicial classifying space of M, i.e. E_{M}T_{•} = BM_{•}. Every diagram F : M^{op} → Spaces comes with a natural transformation π : F → T and hence we get a map of simplicial spaces

E_{M}π_{•} : E_{M}F_{•} → BM_{•}.

The preimage of {i} in the bisimplicial Borel construction is F (i). Denote by E_{M}F the realization ||E_{M}F_{•}||, by BM the realization ||BM_{•}||, and by π_{M} : E_{M}F → BM the map induced by π. We are ready to prove now the main theorem of this section.

Theorem 3.2. [Til97, Theorem 3.2] Let M be a simplicial category and F : M^{op} → Spaces a contravariant diagram. Assume that any morphism f : i → j induces an isomorphism in integral homology H_{∗}(F (j); ℤ) → H_{∗}(F (i); ℤ). Then, for each object i ∈ M, the map F (i) → F ib_{i}(π_{M}) to the homotopy fiber of π_{M} over i is a homology equivalence.

Proof. We apply Theorem 2.4 to the map E_{M}π_{•}. For any n ⩾ 0, the map E_{M}F_{n} → BM_{n} is the projection on the first factors, thus a (homology) fibration. As all faces but d_{0} induce the identity on the fibers, we have only to check that the face map d_{0} induces a homology equivalence on the fibers. Choose a vertex

(f_{n}, …, f_{1}, i_{0}) ∈ mor_{M}(i_{n}, i_{n−1}) × ⋯ × mor_{M}(i_{1}, i_{0}) × {i_{0}}

Its zeroth face is (f_{n}, …, f_{2}, i_{1}) and the map induced on the homotopy fibers is F (f_{0}) : F (i_{0}) → F (i_{1}). This is a homology equivalence by assumption and we are done.

In order to identify the space ΩBM we need to find a diagram F which satisfies the assumptions of Theorem 3.2 and for which the bisimplicial Borel construction E_{M}F is contractible. We give a partial answer to that question which covers the applications made in the context of the mapping class group.

Let us consider for any object j ∈ M the diagram M_{j} as defined in [Til97, Section 3]. It is the restriction of the diagram M, i.e. M_{j}(i) = mor_{M}(i, j). The proof of the next lemma (which we recall here for the sake of completeness) is based on the standard trick to view a simplicial space X_{•} as a vertical simplicial diagram with entries X_{n}. Each of these entries can be seen as a horizontal simplicial diagram with set valued entries X_{n,m}. Commuting the roles of m and n allows to consider X_{•} as a horizontal simplicial space where the space of m-simplices is X_{∗,m}.

Lemma 3.3. [Til97, Lemma 3.3] Let M be a simplicial category and j any object in M. The bisimplicial Borel construction E_{M}M_{j} is then contractible.

Proof. The space of m-simplices of the “horizontal” simplicial space (E_{M}M_{j})_{•} is by definition the nerve of the category of m-simplices of M over the object j. This over-category has a terminal object and is therefore contractible. Hence so is the realization E_{M}M_{j}.

Now fix an object 1 ∈ M and an endomorphism α : 1 → 1, i.e. a vertex in the space of morphisms mor_{M}(1, 1). Form the telescope

M_{∞}(i) = hocolim(M_{1}(i) ⟶^{α_{∗}} M_{1}(i) ⟶^{α_{∗}} …)

Since homotopy colimits commute with themselves E_{M}M_{∞} ≃ hocolimE_{M}M_{1} is contractible and the homotopy fiber of π_{M} is ΩBM. We apply now the theorem to the diagram M_{∞}.

Proposition 3.4. Let M be a simplicial category and assume that there exists an endomorphism α of a specific object 1 such that any morphism f : i → j induces a homology equivalence M_{∞}(j) → M_{∞}(i). Then the natural map M_{∞}(i) → ΩBM is a homology equivalence for any object i ∈ M.

In particular, this means that Bousfield’s homological localization L_{Hℤ} coincides on the spaces M_{∞}(i) and ΩBM (we refer to [Bou75] for the construction and properties of such a localization functor). As the localization L_{Hℤ}M_{∞}(i) is in general difficult to construct, one particularly likes the situation when ΩBM can be identified as Quillen’s plus construction applied to the space M_{∞}(1), because it is better understood. Recall that Quillen’s plus construction X → X^{+} is a homology equivalence which quotients out the maximal perfect subgroup of the fundamental group of X by attaching to X only two- and three-dimensional cells.

We must thus look for conditions ensuring that the map M_{∞}(1) → ΩBM is not only a homology equivalence, but an acyclic map (its homotopy fiber is acyclic). When is this so? In general a homology equivalence is acyclic if the fundamental group of the base space acts nilpotently on the homology of the homotopy fiber (assuming the fiber is connected, see [Ber82, 4.3 (xii)]). This again is usually rather complicated to check, so we are looking for a more convenient condition. Consider the following commutative square in which all arrows are homology equivalences

\xymatrix{M_{∞}(1) \ar[r] \ar[d] & ΩBM \ar[d] \\ M_{∞}(1)^{+} \ar[r] & (ΩBM)^{+}}

Since the fundamental group of any component of a loop space is abelian (ΩBM)^{+} ≃ ΩBM. Moreover a loop space is always nilpotent, hence Hℤ-local. As a homology equivalence between Hℤ-local spaces is an equivalence, we just need a condition that ensures that every connected component of the space M_{∞}(1)^{+} is Hℤ-local as well.

Theorem 3.5. Let M be a simplicial category and assume that there exists an endomorphism α of a specific object 1 such that any morphism f : i → j induces a homology equivalence M_{∞}(j) → M_{∞}(i). Let P be the maximal perfect subgroup of π_{1}M_{∞}(1) and M_{∞}(1)_{P} the corresponding cover. Then we have a weak equivalence M_{∞}(1)^{+} ≃ ΩBM if and only if the quotient π_{1}M_{∞}(1)/P is an abelian group acting trivially on the homology of M_{∞}(1)_{P}.

Proof. Consider the covering fibration M_{∞}(1)_{P} → M_{∞}(1) → K(π_{1}M_{∞}(1)/P, 1), which is quasi-nilpotent, see [Ber82, p.37]. The plus-construction preserves this fibration so we get a new one

M_{∞}(1)_{P}^{+} → M_{∞}(1)^{+} → K(π_{1}M_{∞}(1)/P, 1)

which is quasi-nilpotent again since the action of the fundamental group coincides with the former one. The fundamental group of M_{∞}(1)_{P} is perfect, thus M_{∞}(1)_{P}^{+} is simply-connected. By [Ber82, 4.9] the fibration is nilpotent, i.e. the space M_{∞}(1)^{+} is nilpotent. By Bousfield’s result [Bou75, Theorem 5.5] this means that it is an Hℤ-local space. Therefore the map M_{∞}(1)^{+} → ΩBM is a homology equivalence between Hℤ-local spaces, hence a weak equivalence.

Conversely, if the total space M_{∞}(1)^{+} of the above fibration is a loop space, it is not only Hℤ-local, but it must be a space with an abelian fundamental group acting trivially on all homotopy groups, therefore also on all homology groups.

The stronger condition that π_{1}M_{∞}(1) is perfect is precisely the one checked in the proof of [Til97, Theorem 3.1] to identify the plus construction on the classifying space of the stable mapping class group as a loop space, which turns then out to be an infinite loop space. In [Wah] N. Wahl compares this infinite loop space structure on the stable mapping class group with another one, obtained by operadic means and due to U. Tillmann as well. She proves that both structures actually coincide and makes use of a more amenable variant of Tillmann’s simplicial category.

Remark 3.6. The homology theory which has been used in the present work is integral homology and all applications we know of are obtained working with integral homology. However, with little effort one can replace this homology theory by an arbitrary (possibly extraordinary) homology theory E_{∗}. Hence an E_{∗}-fibration is a map p : E → B such that dp(σ) → F ib_{σ}(p) is an E_{∗}-equivalence. This is equivalent to require that p be a weak E_{∗}-fibration, i.e. dp(σ) → dp(θσ) is an E_{∗}-equivalence for any simplex σ in B and any simplicial operation θ. Then one can prove the analogous of Theorem 2.4: The realization of a natural transformation p_{•} : E_{•} → B_{•} of simplicial spaces where all fibers have the same E_{∗}-homology is an E_{∗}-fibration. The generalized group completion theorem has an E_{∗}-analogue as well, and the question would then be to compare the homotopy type of ΩBM with the E_{∗}- theoretic plus construction.

4. Simplices versus topology

In this section we compare the notions of homology fibration in the category of simplicial sets and topological spaces. So as not to create confusion we will consistently use the terminology “simplicial sets” (not spaces as in the former sections) and emphasize when we deal with topological spaces. The general idea behind simplicial sets is to replace topological data (points) by a combinatorial one (simplices). This is precisely why one defines simplicially a homology fibration by imposing a condition on the preimages of simplices, instead of classically looking at preimages of points. There is however a subtle difference, as shown by the following example due to W. Waldhausen, which we learned from J. Rognes during the BCAT02. A simple map of topological spaces is a map f : X → Y such that the preimages of points f^{−1}(y) ≃ ∗ are contractible for all y ∈ Y. Thus one would be tempted to define simplicially a simple map as a map of simplicial sets f : X → Y for which preimages of simplices dp(σ) ≃ ∗ are all contractible. This is not equivalent to the topological definition. Consider indeed your favorite (but non-trivial) acyclic simplicial set A. The map A → ∗ induces one on the unreduced suspensions ΣA → Δ[1]. The preimage of the simplices in Δ[1] are either points, or ΣA, so all are contractible. But topologically the geometric realization of this map is not simple because the preimage of any other point than the end points of the interval is A.

Recall that a map of topological spaces is a homology fibration if the preimages of all points have the same homology type as the homotopy fiber of p. We prove in this section that the simplicial and topological definitions of homology fibrations are equivalent. Basically this is due to the Mayer–Vietoris Theorem. The idea is to take the barycentric subdivision of the map and reconstruct the preimage of the barycenter of a simplex in the base from the data given by the preimages of the simplices. Let us first recall some standard definitions from [Kan57] (or [FP90, Chapter 4]).

Let μ be a proper face of Δ[n]. We denote by k_{μ} the dimension of μ, that is μ is an injection μ : Δ[k_{μ}] ↪ Δ[n]. The barycentric subdivision of Δ[n], denoted by Δ′[n], is the space which has as q-simplices μ the increasing sequences of q + 1 faces of Δ[n], i.e. μ = (μ_{0}, ⋯, μ_{q}) where μ_{i}(Δ[k_{i}]) ⊂ μ_{i+1}(Δ[k_{i+1}]) for all i ⩽ q − 1. The simplicial operations are the usual: If θ : Δ[q] ⟶ Δ[p] is any simplicial operation then Δ′α(μ) = (μ_{θ(0)}, ⋯, μ_{θ(q)}).

The subdivision functor Sd is left adjoint to Kan’s extension functor Ex (see [Kan57, Section 7]). For any simplicial set E, the q-simplices of SdE are by definition the equivalence classes [x, μ] of a simplex x ∈ E of dimension p and μ ∈ Δ′[p] of dimension q. Two pairs (x, μ) and (x′, μ′) are equivalent if there exists a map α : Δ[p′] → Δ[p] such that x′ = xα and μ = Δ′α(μ). In other words, SdE is the colimit over the simplex category of E of the subdivisions of these simplices: SdE = colim_{ΔE}Δ[n]′.

Let us fix a surjective map f : E → Δ[n]. Its subdivision Sdf : SdE → Δ′[n] is defined as follows. Let [x, μ] be a simplex in SdE as above and consider for any 0 ⩽ i ⩽ q the composite

Δ[k_{i}] ⟶^{μ_{i}} Δ[p] ⟶^{x} E ⟶^{f} Δ[n]

It can be decomposed in a unique way as a degeneracy followed by an injection

Δ[k_{i}] ⟶^{ϕ_{i}} Δ[l_{i}] ↪^{ν_{i}} Δ[n]. Set f([x, μ]) = ν = (ν_{0}, …, ν_{q}).

Definition 4.1. In Δ′[n] fix a vertex α, i.e. a proper face of Δ[n]. The star of α, St(α) is the subspace of Δ′[n] which has as simplices the sequences (μ_{0}, ⋯, μ_{p}) such that ∀i ⩽ p, Imμ_{i} ⊃ Imα. We will further denote by ESt(α) the preimage of St(α) under Sdf.

Lemma 4.2. The inclusion Sdf^{−1}(α) ↪ ESt(α) is a homotopy equivalence.

Proof. Let α be of dimension k. We construct first a retraction r : ESt(α) → Sdf^{−1}(α). Let [x, μ] ∈ ESt(α) be a simplex of dimension q. Then, for any i ⩽ q, there exists a maximal injective morphism Δ[t_{i}] ↪ Δ[k_{i}] (determined by the vertices of μ_{i} whose image under f(x) is a vertex of α) together with a (necessary unique) surjection ϕ : Δ[t_{i}] → Δ[k] rendering the following diagram commutative

\xymatrix{Δ[k_{i}] \ar@{^{(}->}[r]^{μ_{i}} & Δ[p] \ar[r]^{x} & E \ar[d]^{f} \\ Δ[t_{i}] \ar@{^{(}->}[u] \ar[r]^{ϕ} & Δ[k] \ar[r]^{α} & Δ[n]}

We denote the composite Δ[t_{i}] → Δ[k_{i}] → Δ[p] by μ̄_{i} and define r[x, μ] = [x, μ̄]. By construction Sdf([x, μ̄]) is some degeneracy of α. Moreover r is well defined and is clearly a retraction of the inclusion i : Sdf^{−1}(α) ↪ ESt(α).

Finally we construct a homotopy H : ESt(α) × Δ[1] → ESt(α) from i ∘ r to the identity. Let ([x, μ], τ) be a q-simplex in the cylinder, so τ is a q-simplex in Δ[1] and can be represented by a sequence of r + 1 zero’s and q − r one’s: (0 … 01 ⋯ 1). Define then H([x, μ], τ) = [x, μ̄_{0}, …, μ̄_{r}, μ_{r+1}, …, μ_{q}].

In the next proposition we use the decomposition of Δ′[n] as union of all its stars. More precisely consider the category C_{n} whose objects are the non-degenerate simplices of Δ[n] and whose morphisms are generated by the faces σ → d_{i}σ. The unique non-degenerate simplex τ of dimension n is an initial object and diagrams indexed by C_{n} are n-cubes without terminal object. We have Δ′[n] = colim_{σ∈C_{n}}St(σ) = hocolim_{σ∈C_{n}}St(σ) because the diagram St is cofibrant (see for example [DS95]), and even strongly co-Cartesian as defined in [Goo92, Definition 2.1]. Likewise

E ≃ SdE = colim_{σ∈C_{n}}ESt(σ) = hocolim_{σ∈C_{n}}ESt(σ)

Proposition 4.3. Let f : E → Δ[n] be a homology fibration. Then the preimage of the barycenter of Δ′[n] under Sdf has the same homology type as E. In particular the realization |f| : |E| → |Δ[n]| is a homology fibration of topological spaces.

Proof. By Lemma 4.2 the values of the cubical diagram ESt are equivalent to the preimages Sdf^{−1}(σ). When σ is a vertex of Δ[n], one has that Sdf^{−1}(σ) ≃ f^{−1}(σ) = df(σ), which by hypothesis has the same homology type as E. By induction on the dimension of σ we can assume thus that all values in the diagram but the initial one (ESt(τ) ≃ Sdf^{−1}(τ), the preimage of the barycenter) are homology equivalent to E. As the homotopy colimit of the cubical diagram is E, we deduce that ESt(τ) as well has the same homology type as E. We claim that this implies that |f| is a (topological) homology fibration. Indeed by induction again we need only to compute preimages under |f| of points in the interior of the realization of Δ[n]. Any such preimage is a deformation retract of the preimage under |p| of the open simplex, so it is enough to consider the barycenter. The above computation shows precisely that it has the same homology type as |E|, the homotopy fiber of |f|.

Let us now consider a map of simplicial sets p : E → B. To compare both types of homology fibrations we need to control the homological properties of fibers of points in the realization. Any point b ∈ |B| lies in the interior of the realization of a unique non-degenerate simplex σ_{b} ∈ B (see for instance [FP90, Lemma 4.2.5]). Moreover the interior of the realization of σ_{b} embeds in |B|.

Theorem 4.4. A map of simplicial sets p : E → B is a homology fibration if and only if its realization |p| : |E| → |B| is a homology fibration of topological spaces.

Proof. First assume that p : E → B is a homology fibration. We need to compute the homology type of fibers of points in the realization of B and show that the map |p|^{−1}(b) → F ib_{b}(|p|) is a homology equivalence , where F ib_{b}(|p|) denotes the homotopy fiber of |p| over the connected component of b. When σ = σ_{b} is a 0-simplex, this is trivial as p is a homology fibration. If σ is of dimension n ⩾ 1, notice that all the fibers over the points in the interior of |σ| have the same homotopy type (a straightforward computation shows then that the preimage of any point is a deformation retract of the preimage under |p| of the open simplex). Therefore it suffices to analyze the barycenter ι_{n} of the realization of σ and to prove that |p|^{−1}(ι_{n}) → F ib_{ι_{n}}(|p|) is a homology equivalence. As the realization functor commutes with finite limits (see [FP90, Theorem 4.3.16]), we have a pull-back square :

\xymatrix{|dp(σ)| \ar[d] \ar[r] & |E| \ar[d] \\ |Δ[n]| \ar[r]^{σ} & |B|}

The map dp(σ) → Δ[n] is a homology fibration by Proposition 1.3 and its realization is thus a homology fibration by Proposition 4.3: the preimage of the barycenter of |Δ[n]| is homology equivalent to the homotopy fiber |dp(σ)|, which by assumption has the same homology type as the homotopy fiber |F | of |p|.

Assume now |p| : |E| → |B| is a homology equivalence. Inductively we may suppose that for all simplices of dimension ⩽ n − 1 the pull-back dp(τ) is homology equivalent to the homotopy fiber above the component of τ. Let σ be a simplex of dimension n. We have as before a pull-back diagram

\xymatrix{|dp(σ)| \ar[d] \ar[r] & |E| \ar[d] \\ |Δ[n]| \ar[r]^{σ} & |B|}

Decompose dp(σ) as a cubical homotopy colimit dp(σ) ≃ hocolim_{τ∈C_{n}}ESt(τ) following the method seen in the proof of Proposition 4.3. As |p| is a homology fibration, there is a natural transformation by homology equivalences to the constant cubical diagram F ib_{σ}(p) (use Lemma 4.2). A homotopy colimit of homology equivalences is a homology equivalence, hence dp(σ) → F ib_{σ}(p) is a homology equivalence as well.

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Wolfgang Pitsch wpitsch@ima.unige.ch

Institut de Mathématiques, Université de Genève, CH–1200 Genève, Switzerland

Jérôme Scherer jscherer@mat.uab.es

Departament de Matemátiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain