The next goal of this series of posts (started here) is to analyze the oriented cobordism spectrum at the prime 2; the main result is that there is a splitting of into a direct sum of copies of (the torsion-free part) and (the torsion-part). In particular, it will follow that there is only torsion of order two in the cobordism ring — since we showed last time that there was no odd torsion. We will see this using the Adams spectral sequence at the prime , once we’ve figured out what looks like as a comodule over the dual Steenrod algebra. This, however, is apparently somewhat tricky to do directly.

In order to get there, we’ll need a bit of algebraic machinery (which we state in a dual context). Recall that a graded vector space is called *connected* if is one-dimensional and for . The next result provides a sufficient criterion for a module over a graded, connected Hopf algebra to be free.

Theorem 5 (Milnor-Moore)Let be a connected, graded Hopf algebra over a field , and let be a graded, connected -module which is simultaneously a coalgebra(in such a way that is an -homomorphism). Let be a generator, and suppose the map of -modules

is a monomorphism. Then is a free graded -module.

This is a pretty surprising result, as a relatively minor hypothesis (coalgebra, and the action on is free) leads to freeness of the whole thing. The idea of the proof is going to be to produce generators of by lifting a vector space basis of . The fact that these generators are forced to be linearly independent is an unexpected consequence of the coalgebra structure; the graded connectedness will be used to make certain inductive arguments. (more…)