Contents

cohomology

# Contents

## Idea

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).

## Properties

### Existence

###### Proposition

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

$E^\bullet(B U(n)) \;\simeq\; \pi_\bullet(E)[ [ c_1^E, c_2^E, \cdots, c_n^E ] ] \,.$

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)

## References

The concept was introduced in

• P. E. Conner, E. E. Floyd, The relation of cobordism to $K$-theories, Lecture Notes in Mathematics 28 Springer 1966 v+112 pp. MR216511

An early account in a broader context was

More recent textbook and lecture notes include

Some more details are spelled out in

• Riccardo Pedrotti, Complex oriented cohomology, generalized orientation and Thom isomorphism, 2016, 2018 (pdf)

Last revised on November 6, 2018 at 11:35:37. See the history of this page for a list of all contributions to it.