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.