group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The concept of Chern classes of complex vector bundles as universal characteristic classes in ordinary cohomology generalized to any complex oriented generalized cohomology theory: the Conner-Floyd Chern classes (Conner-Floyd 66, Adams 74), review includes (Kochmann 96, section 4.3, Lurie 10, lectures 4 and 5):
for $E$ a generalized cohomology theory, the analog of the first Chern class in $E$-cohomology is what appears in the very definition of complex oriented cohomology. The higher generalized Chern classes are induced from this by the splitting principle. See at complex oriented cohomology – the cohomology ring of BU(n).
The generalized Chern classes serve as the generalized Thom classes that make every complex vector bundle have orientation in generalized cohomology with respect to any complex oriented cohomology theory (Lurie 10, lecture 5, prop. 6).
Given a complex oriented cohomology theory $E$ with complex orientation $c_1^E$, then the $E$-generalized cohomology of the classifying space $B U(n)$ is freely generated over the graded commutative ring $\pi_\bullet(E)$ (prop.) by classes $c_k^E$ for $0 \leq \leq n$ of degree $2k$, these are called the Conner-Floyd-Chern classes
Moreover, pullback along the canonical inclusion $B U(n) \to B U(n+1)$ is the identity on $c_k^E$ for $k \leq n$ and sends $c_{n+1}^E$ to zero.
For $E =$ HZ this reduces to the standard Chern classes.
for details see (Pedrotti 16, prop. 3.1.14)
The concept was introduced in
An early account in a broader context was
More recent textbook and lecture notes include
Stanley Kochmann, section 4.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Jacob Lurie, Chromatic Homotopy Theory, 2010, lecture 4 (pdf) and lecture 5 (pdf)
Some more details are spelled out in
Last revised on November 6, 2018 at 11:35:37. See the history of this page for a list of all contributions to it.