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 -category. Dror didn’t have this language available, but his results fit neatly into it.
Let be the -category of pointed spaces. We have a functor
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 is . We can look at the subcategory consisting of spaces sent by to zero (that is, to a contractible complex).
Definition 1 Spaces in are called acyclic spaces.
The subcategory is closed under colimits (as is colimit-preserving). It is in fact a very good candidate for a homotopy theory: that is, it is a presentable -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.
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 , the complement has trivial homology but is likely non-contractible.
There are also a number of acyclic groups with the property that . These groups can never be finite: for finite , there is a spectral sequence (the Atiyah-Hirzebruch spectral sequence)
where is the completion of the representation ring (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.
The functor is colimit-preserving, in the -categorical sense. The adjoint functor theorem implies that there is a right adjoint
called the acyclicization functor. It behaves somewhat dually to the theory of Bousfied localization. For any pointed space , we have a natural map
which exhibits as the “maximal” acyclic space mapping to . (More precisely, is universal with respect to maps into from an acyclic space.)
Dror’s paper describes very explicitly via an inductive process of killing homology groups. Namely, Dror defines
where the are “approximations,” spaces over defined recursively:
- is the covering space corresponding to the unique maximal perfect subgroup of . (We throw away all connected components of ignoring the basepoint.) In other words, it is the homotopy fiber of
where is the maximal perfect subgroup.
- Given which is assumed to be -acyclic (that is, for ), we let be the homotopy fiber of
Dror’s first main result is:
Theorem 2 (Dror) Each satisfies for , and
In particular, , which was described very abstractly through the adjoint functor theorem (equivalently, as a colimit over all acyclic spaces mapping to ) now admits a slightly more manageable description: there is an algorithmic process, given , to make it acyclic.
Proof: Let’s first see that each is -acyclic. When , it follows from the Hurewicz theorem, since
is the maximal perfect subgroup. If we knew that is -acyclic, then an argument with the Serre spectral sequence shows that is -acyclic. The fiber sequence
and the fact that
is an isomorphism imply this fact: run the spectral sequence converging to the homology of . This looks like
Let’s take , for example. Since is 3-connected, we know that the first homology of can occur in dimension . So the spectral sequence looks like:
Looking at the diagonal line , we observe that the term accounts for all the homology in ; therefore, the term must be zero. More generally, is -acyclic.
It’s a consequence of this that is itself acyclic. In general, homology does not commute with homotopy inverse limits, but in this case the converge “well” to : that is, is highly connected for large. The partial acyclicity of the thus imply the acyclicity of .
Finally, we need to see that for any acyclic space , the map
is a homotopy equivalence (where the ‘s refer to mapping spaces). To see this, it suffices to see that is a homotopy equivalence for each . When , we note that a map from an acyclic space into lifts uniquely to the cover of , because an acyclic space’s is necessarily perfect.
For higher , we have a fiber sequence
for some abelian group . The last space is contractible for acyclic , which means that as far as acyclic spaces are concerned, one can’t see the difference between and . Taking inverse limits, we get that is a homotopy equivalence.
Dror’s model for acyclic spaces does more: it provides us with theorems about the acyclization functor. For instance, we can see that if is -acyclic to begin with, then the map
is a -connected, since we obtained by taking homotopy fibers of maps into Eilenberg-MacLane spaces of degree . So if is “close” to being acyclic, then modifies “very little.”
A related observation is that the functor preserves filtered colimits. Namely, Dror’s model shows that is constructed as a homotopy inverse limit of functors . Each is constructed inductively by forming a homotopy pull-back using , and we can thus see inductively that each commutes with filtered colimits. The functor is obtained by an infinite homotopy limit from the , which does not obviously commute with filtered colimits; however, we recall that is -connected for each . The “strong” convergence of to implies (e.g. by inspecting homotopy groups) that itself commutes with filtered colimits.