algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Conner-Floyd isomorphism (Conner-Floyd 66, Thm. 10.1, Conner-Smith 69, Thm. 9.1) is a natural isomorphism
between the complex topological K-theory group of a finite CW-complex and the extension of scalars of the MU-cobordism cohomology of along the Todd genus (where is the MU-cobordism ring of stably almost complex manifolds , and is their Todd class).
A slightly more abstract way of saying the same is
which – thinking now of the Todd genus as coming from the canonical complex orientation (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).
The original articles on the cases MUKU and MSpKO:
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 MUKU in:
Review:
See also:
Gerhard Wolff, Der Einfluss von auf , 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 MSUKO is due to:
The version for MSpin^cMU and MSpinKO 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.