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.