nLab
Conner-Floyd Chern class

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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 EE a generalized cohomology theory, the analog of the first Chern class in EE-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 EE with complex orientation c 1 Ec_1^E, then the EE-generalized cohomology of the classifying space BU(n)B U(n) is freely generated over the graded commutative ring π (E)\pi_\bullet(E) (prop.) by classes c k Ec_k^E for 0n0 \leq \leq n of degree 2k2k, these are called the Conner-Floyd-Chern classes

E (BU(n))π (E)[[c 1 E,c 2 E,,c n E]]. 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 BU(n)BU(n+1)B U(n) \to B U(n+1) is the identity on c k Ec_k^E for knk \leq n and sends c n+1 Ec_{n+1}^E to zero.

For E=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 KK-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.