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.
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 , 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.
2. Acyclicization
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.
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