This is the third in a series of posts on oriented cobordism. In the first post, we analyzed the spectrum at odd primes; in this post, we will analyze the prime . After this, we’ll be able to deduce various classical geometric facts about manifolds.

The next goal is to determine the structure of the homology as a comodule over . Alternatively, we can determine the structure of the *cohomology* over the Steenrod algebra : this is a coalgebra and a module.

Theorem 8 (Wall)As a graded -module, is a direct sum of shifts of copies of and .

This corresponds, in fact, to a *splitting* at the prime 2 of into a wedge of Eilenberg-MacLane spectra.

In fact, this will follow from the comodule structure theorem of the previous post once we can show that if is the Thom class, then the action of on has kernel : that is, the only way a cohomology operation can annihilate if it is a product of something with . Alternatively, we have to show that the complementary Serre-Cartan monomials in applied to ,

are linearly independent in . (more…)