nLab Conner-Floyd isomorphism

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

• for MSp and KO (Conner-Floyd 66)

and

• for MSpin and KO as well as MSpin^c and KU under the Atiyah-Bott-Shapiro orientation (Hopkins-Hovey 92, Thm. 1)

However, the analogous statement for

• for MSU and KO

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

Related concepts

• Landweber exact functor theorem

• Conner-Floyd Chern class

References

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

• Pierre Conner, Edwin Floyd, Theorem 10.1 in: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

  
  

  Russian transl.: Коннер П., Флойд Э. 0 соотношении теории кобордизмов и К-теории. - Дополнение к кн.: Гладкие периодические отображения. - М.: Мир, 1969 (djvu)

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

• Pierre Conner, Larry Smith, Theorem 9.1 in: On the complex bordism of finite complexes, Publications Mathématiques de l’IHÉS, Tome 37 (1969) , pp. 117-221 (numdam:PMIHES_1969__37__117_0)

Review:

• Dai Tamaki, Akira Kono, Theorem 6.35 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

See also:

• 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:

• Serge Ochanine, Modules de SU-bordisme. Applications, Bulletin de la Société Mathématique de France, Tome 115 (1987) , pp. 257-289 (doi:10.24033/bsmf.2078)

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

• Michael Hopkins, Mark Hovey, Spin cobordism determines real K-theory, Mathematische Zeitschrift volume 210, pages 181–196 (1992) (doi:10.1007/BF02571790, pdf)

Generalization to equivariant cohomology theory, now relating equivariant cobordism cohomology to equivariant K-theory:

• Steven Costenoble, Equivariant Conner-Floyd isomorphism, Trans. Amer. Math. Soc. 304 2 (1987) 801–818, [jstor:2000742]

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

On a construction of algebraic cobordism as a non-$\mathbb{A}^1$-invariant cohomology theory on derived schemes, and the resulting Conner-Floyd isomorphism:

• Toni Annala, Marc Hoyois, Ryomei Iwasa, Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory [arXiv:2303.02051]