nLab homology of MG

Idea

The (generalised) homology of Thom spectra $M G$, 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 $B G$, such as $B O$ and $B U$.

Feeding the homology of $M G$ 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 $MU$.

Statement

Proposition

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

$$E_\bullet(M U) \simeq (\pi_\bullet E)[b_1, b_2, \cdots] \,,$$

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

(Lurie 10, lecture 7, prop. 2)

Related concepts

• Boardman homomorphism

• homology of MO

References

• Frank Adams, part II.6 of Stable homotopy and generalised homology, 1974

• Stanley Kochmann, section 2.6 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Jacob Lurie, lecture 7 of Chromatic Homotopy Theory, 2010, (pdf)