nLab
Conner-Floyd isomorphism

Contents

Context

Algebraic topology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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

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

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

A slightly more abstract way of saying the same is

KU (X)MU (X) MU KU 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 MUKUMU \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\toKU and MSp\toKO:

with an alternative proof for MU\toKU in:

Review:

See also:

  • Gerhard Wolff, Der Einfluss von K *()K^{\ast} (-) auf U *()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\toKO is due to:

The version for MSpin^c\toMU and MSpin\toKO 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.