group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 $MU$.
For $E$ a complex oriented cohomology theory, there is an isomorphism of graded rings
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)
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)
Last revised on March 5, 2024 at 00:03:59. See the history of this page for a list of all contributions to it.