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 1For 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. *