May 24, 2012

The homology of MU

Posted by Akhil Mathew under algebra, topology | Tags: , , , |
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Finally, it’s time to try to understand the computation of the cobordism ring {\pi_* MU}. This will be the first step in understanding Quillen’s theorem, that the formal group law associated to {MU} is the universal one. We will compute {\pi_* MU} using the Adams spectral sequence.

In this post, I’ll set up what we need for the Adams spectral sequence, which is a little bit of algebraic computation. In the next post, I’ll describe the actual calculation of the spectral sequence, which will complete the description of \pi_* MU.

1. The homology of {MU}

The starting point for all this is, of course, the homology {H_*(MU)}, which is a ring since {MU} is a ring spectrum. (In the past, I had written reduced homology {\widetilde{H}_*(MU)} for spectra, but I will omit it now; recall that for a space {X}, we have {\widetilde{H}_*(X) = H_*(\Sigma^\infty X)}.)

Anyway, let’s actually do something more general: let {E} be a complex-oriented spectrum (which gives rise to a homology theory). We will compute {E_*(MU) = \pi_* E \wedge MU}.

Proposition 1 {E_*(MU) = \pi_* E [b_1, b_2,\dots]} where each {b_i} has degree {2i}.

The proof of this will be analogous to the computation of {H_* (MO; \mathbb{Z}/2)}. In fact, the idea is essentially that, by the Thom isomorphism theorem,

\displaystyle E_*(MU) = E_*(BU) \simeq \pi_* E [b_1, \dots, ]

where the last equality is because {E} is complex-oriented, and consequently the {E}-homology of {BU} looks like the ordinary homology of it. (more…)

 

May 22, 2012

Chromatic homotopy theory II: the universal complex-oriented theory

Posted by Akhil Mathew under topology | Tags: , , |
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(This is the second in a series of posts intended for me to try to understand the connection between stable homotopy theory and formal group laws.)

Last time, we introduced the notion of a complex-oriented cohomology theory and saw that we could imitate the classical theory of Chern classes in one such. In this post, I’d like to describe the universal example of a complex-oriented cohomology theory: complex cobordism. This is going to play a very special role in the next few posts.

To unravel this, let’s try to recall what a complex orientation was. It was a choice of Thom classes of complex vector bundles, functorial in the bundle and multiplicative. For starters, let’s focus now on the functoriality. Let {E} be a cohomology theory represented by a spectrum {E}. Then since there is a universal {n}-dimensional vector bundle {\zeta_n \rightarrow BU(n)}, it follows that a functorial choice of Thom classes for {n}-dimensional vector bundles is the same as a Thom class for {\zeta_n}. So, all we need to give is an element of {\widetilde{E}^*(T(\zeta_n))}. If we (and we henceforth do this) normalize things such that the Thom class of the {n}-th degree element is in degree {n}, then we have to give an element of {\widetilde{E}^n(T(\zeta_n))}.

Definition 1 The spectrum {MU(n)} is {\Sigma^{-2n}T(\zeta_n)}.

So another way of saying this is that we should have a map of spectra

\displaystyle MU(n) \rightarrow E.

There is a map {S^{2n} \rightarrow T(\zeta_n)} which comes from fixing a basepoint in {BU(n)}. So, in other words, to give a functorial complex orientation for {n}-dimensional complex vector bundles is to give an element of {\widetilde{E}^n(T(\zeta_n))} which restricts to the generator of {\widetilde{E}^n(S^{2n})}. (To check that an element is a Thom class, we only need restrict it to one fiber in each connected component of the base.) (more…)