Contents

cohomology

# Contents

## Idea

The Conner-Floyd isomorphism (Conner-Floyd 66, Thm. 10.1, Conner-Smith 69, Thm. 9.1) is a natural isomorphism

$KU^\bullet(X) \cong MU^\bullet(X) \otimes_{\Omega^U} \mathbb{Z}$

between the complex topological K-theory group of a finite CW-complex $X$ and the extension of scalars of the MU-cobordism cohomology of $X$ along the Todd genus $\Omega^U \overset{Td}{\longrightarrow} \mathbb{Z}$ (where $\Omega^U_{-\bullet} = MU^\bullet(\ast)$ is the MU-cobordism ring of stably almost complex manifolds $\Sigma$, and $Td(\Sigma) \in \mathbb{Z}$ is their Todd class).

A slightly more abstract way of saying the same is

$KU^\bullet(X) \cong MU^\bullet(X) \otimes_{MU^\bullet} KU^\bullet\,$

which – thinking now of the Todd genus as coming from the canonical complex orientation $MU \longrightarrow KU$ (see at universal complex orientation of MU) – shows that the Conner-Floyd isomorphism is a special case of the Landweber exact functor theorem.

The analogous statement holds

and

However, the analogous statement for

(via the Conner-Floyd orientation) fails, or rather does hold with a small modification (Ochanine 87).

## References

The original articles on the cases MU$\to$KU and MSp$\to$KO:

with an alternative proof for MU$\to$KU in:

Review:

• Gerhard Wolff, Der Einfluss von $K^{\ast} (-)$ auf $U^{\ast} (-)$, Manuscripta Math. 10 (1973), 141–-161 (doi:10.1007/BF01475039)

• Gerhard Wolff, Vom Conner-Floyd Theorem zum Hattori-Stong Theorem, Manuscripta Math. 17 (1975), no. 4, 327–-332 (doi:10.1007/BF01170729, MR388420)

The (failure of the) version for MSU$\to$KO is due to:

The version for MSpin^c$\to$MU and MSpin$\to$KO is due to:

Generalization to equivariant cohomology theory:

Discussion in motivic cohomology:

• David Gepner, Victor Snaith, On the motivic spectra representing algebraic cobordism and algebraic K-theory, Doc. Math., 14:359–396 (electronic), 2009, pdf

• Ivan Panin, Konstantin Pimenov, Oliver Röndings, On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory, Invent. Math., 175 (2009), no. 2, 435–451., MR2470112

Last revised on February 18, 2021 at 11:37:17. See the history of this page for a list of all contributions to it.