In the previous post, we were trying to show that any homology class of a space in dimension at most six can be represented by a smooth oriented manifold mapping to . This statement is a geometric one, but it can be proved via homotopy-theoretic means. In the previous post, we interpreted it in terms of homotopy theory, and we showed that
was a surjection in degrees (actually, in degrees ) for either or an odd prime . In this post, we will handle the case . Namely, we will produce an approximation to in the first few homotopy groups (essentially, we’ll work out the first couple of pieces of a Postnikov decomposition). This will give a criterion for when a homology class in low degrees is in the image of , and we’ll see that it is always satisfied in degrees . This will complete the proof of:
Theorem 1 For any space , the map is surjective for : that is, any homology class of dimension is representable by a smooth manifold.
In the case of an odd prime , we used as a 7-approximation to . This is not going to work at , because the cohomologies are off. Namely, the cohomology of at has two generators in degrees (namely, the Thom class and for the first Pontryagin class). However, has four generators in mod cohomology in these dimensions: for the tautological classes. So the Postnikov decomposition is going to look somewhat different.
Most of this material described in the past few posts comes from a variety of sources: Thom’s original paper (Quelques propriétés globales), Rudyak’s On Thom Spectra, Orientability, and Cobordism, and Stong’s Notes on Cobordism Theory.
1. The Postnikov decomposition of
Instead, consider the cohomology operation which is given by the composite of the Bockstein and . In other words, this cohomology operation on a space acts as
( is a variant of the “Steenrod cube” obtained by doing the same with mod 2 instead of .) Let be the homotopy fiber of , so that we have an exact triangle
from which it follows that is a “twisted” product of the Eilenberg-MacLane spectra . However, we observe that the cohomology of in degrees has only two generators, one in degree one (the pullback of ) and one in degree four (). So, we might expect to be a better 3-local approximation to in low degrees.
Let’s now work entirely 3-locally: will be regarded as a map . The map
is zero (in fact, we’ve seen that is zero in dimension five), so we can find a lift .
Proposition 2 is a 7-equivalence.
Proof: As before, we just have to prove that it is a 6-equivalence, since the homotopy groups of both spaces vanish in dimension . So we might as well prove that the map on cohomology
is an equivalence in dimensions .
There is a commutative diagram:
Now, as we saw, is generated by the pull-back and . These classes pull back in to the Thom class and . Since we know that the cohomology of in these dimensions is generated by and , it suffices to show that
This is a general fact that holds on the Thom space of an oriented vector bundle. In other words, we have:
Lemma 3 Let be an oriented vector bundle, and let be the Thom class. Then .
This is analogous to the statement with and , for instance.
By the properties of both and , we find that if the result is true for vector bundles , then it is true for . Using the “splitting principle,” we can reduce to an oriented 2-dimensional bundle, or what is the same thing, a complex line bundle. So it suffices to prove the above lemma in the “universal” complex line bundle case: that is, when one regards as the Thom space of the over .
In this case, if is the hyperplane class, then it is also the Thom class, and . So
which verifies the claim.
With the lemma proved, the proposition is proved as well.
2. Completion of the proof
Now suppose is a space. We have a map
which is an isomorphism for and a surjection for , because the cofiber of has homotopy groups concentrated in dimension and above (and thus so does its smash product with ). This is good, but we do not know that surjects onto . Rather, there is an exact sequence for each
So, the upshot of all this is that an element in is hit by an element of if and only if the dual homology operation
annihilates it. We find as a result:
Proposition 4 For , a homology class in is in the image of if and only if it is annihilated by the dual homology operation .
Our goal is now to show that is always zero in degrees . In dimensions , this is evident, since decreases degree by . Let us now handle dimension .
Proposition 5 For any space , acts as the zero operation on .
Proof: In fact, is the composite of the Bockstein and the dual to , which will be written as (this decreases homology degree by four). Namely, it is the composite:
where is the homology version of the Bockstein.
When , the relevant Bockstein is going from to . But this Bockstein is always zero: that is, consider the exact sequence
Since multiplication by is injective on , the Bockstein is forced to be zero.
Slightly more subtle is the next fact, which will complete the proof of the theorem:
Proposition 6 For any finite CW complex , acts as the zero operation on .
Proof: This is a little trickier. The strategy is to pick a class . If is not zero, we can evaluate on a class in and get a nonzero element of if is appropriate; this is because is a finitely generated -module. Consequently, for every nonzero element of , we can choose a map
for some , which does not annihilate it.
Choose such a class . This amounts to pairing with , so we can say that . Now, is pulled back from the tautological class in via a map . In other words, we find that
In other words, we are reduced by naturality to the “universal” case of a . But , so the proposition is trivial in this case. This can be proved as follows: is the sixth group homology of the cyclic group , and one can compute this by an explicit resolution to see that it is zero in positive even dimensions (see for instance this page).
This, finally, completes the proof that homology classes in degrees are representable by smooth manifolds. Note that we also get a necessary and sufficient condition in dimension seven:
Corollary 7 A homology class is representable by a smooth manifold if and only if .