I recently read E. Dror’s paper “Acyclic spaces,” which studies the category of spaces with vanishing homology groups. It turns out that this category has a fair bit of structure; in particular, it has a theory resembling the theory of Postnikov systems. In this post and the next, I’d like to explain how the results in Dror’s paper show that the decomposition is really a special case of the notion of a Postnikov system, valid in a general {\infty}-category. Dror didn’t have this language available, but his results fit neatly into it.

Let {\mathcal{S}} be the {\infty}-category of pointed spaces. We have a functor

\displaystyle \widetilde{C}_*: \mathcal{S} \rightarrow D( \mathrm{Ab})

into the derived category of abelian groups, which sends a pointed space into the reduced chain complex. This functor preserves colimits, and it is in fact uniquely determined by this condition and the fact that {\widetilde{C}_*(S^0)} is {\mathbb{Z}[0]}. We can look at the subcategory {\mathcal{AC} \subset \mathcal{S}} consisting of spaces sent by {\widetilde{C}_*} to zero (that is, to a contractible complex).

 

Definition 1 Spaces in {\mathcal{AC}} are called acyclic spaces.

 

The subcategory {\mathcal{AC} \subset \mathcal{S}} is closed under colimits (as {\widetilde{C}_*} is colimit-preserving). It is in fact a very good candidate for a homotopy theory: that is, it is a presentable {\infty}-category. In other words, it is a homotopy theory that one might expect to describe by a sufficiently nice model category. I am not familiar with the details, but I believe that the process of right Bousfield localization (with respect to the class of acyclic spaces), can be used to construct such a model category.

 

1. Examples

One thing to check is that this theory is nontrivial: that is, there are non-contractible examples of acyclic spaces. One source of examples comes from homology spheres, or manifolds with the homology (but not necessarily the homotopy!) of a sphere, such as the one due to Poincaré. Given a homology sphere {M}, the complement {M \setminus \left\{\ast\right\}} has trivial homology but is likely non-contractible.

There are also a number of acyclic groups {G} with the property that {H^*(BG; \mathbb{Z}) = 0}. These groups can never be finite: for finite {G}, there is a spectral sequence (the Atiyah-Hirzebruch spectral sequence)

\displaystyle H^*(BG; \mathbb{Z}) \rightarrow K^*(BG),

where {K^*(BG)} is the completion of the representation ring {R(G)} (by the Atiyah-Segal completion theorem) and is never zero. However, very large groups can be acyclic. A discussion of classical examples may appear in a later post.

2. Acyclicization

The functor {\mathcal{AC} \subset \mathcal{S}} is colimit-preserving, in the {\infty}-categorical sense. The adjoint functor theorem implies that there is a right adjoint

\displaystyle A: \mathcal{S} \rightarrow \mathcal{AC},

called the acyclicization functor. It behaves somewhat dually to the theory of Bousfied localization. For any pointed space {X}, we have a natural map

\displaystyle A X \rightarrow X ,

which exhibits {AX} as the “maximal” acyclic space mapping to {X}. (More precisely, {AX \rightarrow X} is universal with respect to maps into {X} from an acyclic space.)

Dror’s paper describes {AX} very explicitly via an inductive process of killing homology groups. Namely, Dror defines

\displaystyle AX = \varprojlim A_n X,

where the {A_n X} are “approximations,” spaces over {X} defined recursively:

  1. {A_1 X \rightarrow X} is the covering space corresponding to the unique maximal perfect subgroup of {\pi_1(X)}. (We throw away all connected components of {X} ignoring the basepoint.) In other words, it is the homotopy fiber of

    \displaystyle X \rightarrow B (\pi_1 X / C),

    where {C \subset \pi_1 X} is the maximal perfect subgroup.

  2. Given {A_{n-1} X} which is assumed to be {(n-1)}-acyclic (that is, {\widetilde{H}_i(A_{n-1} X) = 0} for {i \leq n-1}), we let {A_n X} be the homotopy fiber of

    \displaystyle A_{n-1} X \rightarrow K( H_n(A_{n-1} X); n).

Dror’s first main result is:

 

Theorem 2 (Dror) Each {A_{n} X} satisfies {\widetilde{H}_i(A_n X) = 0} for {i \leq n}, and

\displaystyle AX = \varprojlim A_n X.

 

In particular, {AX}, which was described very abstractly through the adjoint functor theorem (equivalently, as a colimit over all acyclic spaces mapping to {X}) now admits a slightly more manageable description: there is an algorithmic process, given {X}, to make it acyclic.

Proof: Let’s first see that each {A_n X} is {n}-acyclic. When {n = 1}, it follows from the Hurewicz theorem, since

\displaystyle \pi_1 A_1 X \subset \pi_1 X

is the maximal perfect subgroup. If we knew that {A_{n-1} X} is {(n-1)}-acyclic, then an argument with the Serre spectral sequence shows that {A_n X} is {n}-acyclic. The fiber sequence

\displaystyle A_n X \rightarrow A_{n-1} X \rightarrow K( H_n(A_{n-1} X); n)

and the fact that

\displaystyle H_n(A_{n-1} X) \rightarrow H_n ( K( H_n(A_{n-1} X); n) )

is an isomorphism imply this fact: run the spectral sequence converging to the homology of {A_{n-1} X}. This looks like

\displaystyle H_p( K(H_n( A_{n-1}, n)); H_q( A_n X)) \implies H_{p+q}(A_{n-1} X).

Let’s take {n = 4}, for example. Since {A_4 X \rightarrow A_3 X} is 3-connected, we know that the first homology of {A_4 X} can occur in dimension {4}. So the spectral sequence looks like:

Looking at the diagonal line {p+q = 4}, we observe that the term {H_4( A_3 X)} accounts for all the homology in {H_4(A_3 X)}; therefore, the term {H_4( A_4 X)} must be zero. More generally, {A_n X} is {n}-acyclic.

It’s a consequence of this that {AX} is itself acyclic. In general, homology does not commute with homotopy inverse limits, but in this case the {A_n X} converge “well” to {AX}: that is, {AX \rightarrow A_n X} is highly connected for {n} large. The partial acyclicity of the {A_n X} thus imply the acyclicity of {AX}.

Finally, we need to see that for any acyclic space {Y}, the map

\displaystyle \hom(Y, AX) \rightarrow \hom(Y, X)

is a homotopy equivalence (where the {\hom}‘s refer to mapping spaces). To see this, it suffices to see that {\hom(Y, A_n X) \rightarrow \hom(Y, A_{n-1} X)} is a homotopy equivalence for each {n}. When {n = 1}, we note that a map from an acyclic space into {X} lifts uniquely to the cover {A_1 X} of {X}, because an acyclic space’s {\pi_1} is necessarily perfect.

For higher {n}, we have a fiber sequence

\displaystyle \hom(Y, A_n X) \rightarrow \hom(Y, A_{n-1} X) \rightarrow \hom(Y, K(G, n)),

for some abelian group {G}. The last space is contractible for acyclic {Y}, which means that as far as acyclic spaces are concerned, one can’t see the difference between {A_n X} and {A_{n-1}X}. Taking inverse limits, we get that {\hom(Y, AX) \rightarrow \hom(Y, X)} is a homotopy equivalence. \Box

Dror’s model for acyclic spaces does more: it provides us with theorems about the acyclization functor. For instance, we can see that if {X} is {n}-acyclic to begin with, then the map

\displaystyle A X \rightarrow X

is a {n}-connected, since we obtained {AX } by taking homotopy fibers of maps into Eilenberg-MacLane spaces of degree {>n}. So if {X} is “close” to being acyclic, then {A} modifies {X} “very little.”

A related observation is that the functor {A} preserves filtered colimits. Namely, Dror’s model shows that {A} is constructed as a homotopy inverse limit of functors {A_n }. Each {A_n} is constructed inductively by forming a homotopy pull-back using {A_{n-1}}, and we can thus see inductively that each {A_n} commutes with filtered colimits. The functor {A} is obtained by an infinite homotopy limit from the {A_n}, which does not obviously commute with filtered colimits; however, we recall that {A \rightarrow A_n} is {n}-connected for each {n}. The “strong” convergence of {A_n} to {A} implies (e.g. by inspecting homotopy groups) that {A} itself commutes with filtered colimits.

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