Update: (9/25) I just found a nice paper by Andrew Ranicki explaining the algebraic interpretation of the finiteness obstruction.
This is the second piece of a two-part post trying to understand some of the ideas in Wall’s “Finiteness conditions for CW complexes.”
In the previous post, we considered a space which was a homotopy retract of an -dimensional finite CW complex (where ), and tried to express itself as homotopy equivalent to one such. We built a sequence of approximations
of complexes over , such that each was an -dimensional finite complex and such that for : the maps increase in connectivity at each stage. In general, we cannot make this sequence stop. However, we saw that if was chosen such that the -module
was free (where the tilde denotes the universal cover), then we could build from (by attaching -cells) in such a way that was a homotopy equivalence: that is, and .
The goal now is to use this requirement of freeness to build a finiteness obstruction in analogy with the algebraic situation considered in the previous post. Namely, let be any connected space. Then the universal cover is a -space, and the singular chain complex is a complex of -modules: that is, it lives in the derived category of -modules. We will see below that if is a finite complex, then it lives in the “finitely presented” derived category introduced in the previous post—so that if is finitely dominated, then is in the perfect derived category of .
Definition 1 The Wall finiteness obstruction of is the class in represented by the complex : that is, choose a finite complex of finitely generated projective modules representing , and take .
It follows from the assertions made in the previous paragraph, which are proved below, that the vanishing of the finiteness obstruction is necessary for to live in the finitely presented derived category, and for to be a finite cell complex. Sufficiently will take a little extra work, using the constructions of the previous post. We’ll also get to see that the above definition recovers the output of those constructions.
In order for this definition to make sense, there are a number of lemmas to check. For one thing, we need to show that if is a finitely dominated space, then lives in the perfect derived category of . This will follow from the next lemma, which has already been alluded to a couple of times.
Lemma 1 Let be a discrete group. Let be a finite complex with a map , classifying a principal -bundle . Then (a -module) lives in the finitely presented derived category of -modules.
Proof: In fact, the category of finite complexes over is generated under finite (homotopy) colimits by the map . The association is the composite of two functors (pull-back and ) each of which preserves homotopy colimits. In particular, is always in the smallest subcategory of closed under finite colimits containing the image under the above functor of . But the image of corresponds to the chain complex of the discrete -set , which is concentrated in degree zero: this complex generates precisely the finitely presented derived category.
So if is any finite connected complex, there is a map whose (homotopy) fiber is the principal bundle . The above lemma shows that is in the finitely presented derived category . For a retract of a finite complex, it follows that is simply a perfect complex. In particular, we find:
Theorem 2 (Necessity) If the Wall finiteness obstruction in is nonzero, then is not of the homotopy type of a finite complex.
Let’s now suppose that is finitely dominated and that its Wall finiteness obstruction vanishes. We’d like to show that it is in fact a finite complex. In order to do this, let’s combine the algebra with the homotopy-theoretic constructions of the previous post. Choose such that for and all groups , and moreover such that for all coefficient systems on . We can do this either by noting the finite-dimensionality of the dominating space, or by using the fact that is a perfect -complex.
As in the previous post or the beginning of this one, we construct a sequence of successive approximations
where the one we care about is the last one. That is, is a finite -dimensional complex with the property that
We saw in the previous post that freeness of the -module
would allow us to choose a (by attaching finitely many -cells) homotopy equivalent to .
However, let’s consider the chain complexes and ; these live in the perfect derived category of . There is a natural cofiber sequence
We note that has no homology in indices as in these dimensions, and also for indices because is -dimensional. In particular, we have an equivalence (in the derived category)
Observe moreover that is a perfect complex, and that its -theory invariant is the opposite to that of (because that of vanishes).
Lemma 3 The -module is a finitely generated projective.
Proof: In order to see this, we need to see that maps in the derived category from into , for any -module , are zero, as these are the extension groups which we’d like to vanish. However, these are precisely maps
or since has no (possibly twisted) cohomology above level , maps . These measure the (twisted) cohomology , which we have assumed vanishes.
In particular, we find that the Wall finiteness obstruction is, up to a sign, equal to the class of the projective module in -theory. If the finiteness obstruction vanishes, then this -module is stably free. In this case, we can modify to build by first adding a bunch of -spheres (which map trivially to ) so as to make a new -dimensional complex . Then for . Moreover, is obtained from , as a -module, by adding free summands, one for each added.
If we add enough of these, we can arrange it so that is actually free. Then, we are in the previous situation: we already know to add a bunch of -cells to to construct a mapping to via a homotopy equivalence.
Theorem 4 If is finitely dominated and the Wall finiteness obstruction vanishes, then is homotopy equivalent to a finite CW complex.
Let’s now show that the Wall finiteness obstruction can be nonzero. More precisely, let’s show that given any finitely presented group , and any class in , we can construct a complex with equal to and with finiteness obstruction equal to this class.
Let be a finitely generated projective -module which is not stably free. By definition, is the image of an idempotent for some . Such an idempotent corresponds to the choice of .
Start by choosing a finite complex with . Choose , and consider the complex . We note that
as -modules. In particular, we can choose maps corresponding to the generators , and wedging with , they define a map
such that is given by the action of in .
the claim is that the finiteness obstruction of is given by (up to a sign depending on the parity of ), the class of . In fact, we note that has the property that is quasi-isomorphic to , by construction.
Moreover, is finitely dominated (we should have checked this first): the map admits a section up to homotopy. To see this, we can convert into a fibration. The obstructions to finding a section are in the groups . For , this is automatically zero since has no cohomology above . For , the relevant homotopy groups are zero.
Note in particular that these -groups can be very nonzero: if for instance , the relevant -groups are those of the Dedekind domain and in particular are the class groups of cyclotomic fields, which are generally nontrivial.