nLab
homology of MG

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The (generalised) homology of Thom spectra MGM G, such as MO and MU.

In the presence of a Thom isomorphism (e.g. for complex oriented cohomology of MU) this identifies with the homology of the classifying spaces BGB G, such as BOB O and BUB U.

Feeding the homology of MGM G into an Adams spectral sequence gives a way to compute its homotopy groups.

This is for instance a key ingredient in Quillen's theorem on MU giving the homotopy groups of MUMU.

Statement

Proposition

For EE a complex oriented cohomology theory, there is an isomorphism of graded rings

E (MU)(π E)[b 1,b 2,], E_\bullet(M U) \simeq (\pi_\bullet E)[b_1, b_2, \cdots] \,,

where the {b iE (MU(1))}\{b_i \in E_\bullet(M U(1))\} are a dual basis to the basis {t i+1E (MU)t(π E)[[t]]}\{t^{i+1} \in E^\bullet(M U) \simeq t (\pi_\bullet E)[ [ t ] ]\} that is induced by the complex orientation.

(Lurie 10, lecture 7, prop. 2)

References

Last revised on February 17, 2016 at 04:25:08. See the history of this page for a list of all contributions to it.