group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A K-orientation is an orientation in generalized cohomology for K-theory (typically: topological K-theory).
For ordinary topological K-theory a smooth function between smooth manifolds $f \colon X \to Y$ is K-oriented if $T X \oplus f^\ast(T Y)$ has a spin^c structure.
In this case there is an Umkehr map
There is a universal orientation in generalized cohomology of
complex K-theory$\,$KU over spin^c structures;
KO over spin structure
hence E-infinity ring homomorphisms out of the Thom spectrum MSpin
and
These are (both) referred to as the Atiyah-Bott-Shapiro orientation (after Atiyah-Bott-Shapiro 64); the $E_\infty$-structure is due to (Joachim 04).
The genus induced by $M Spin \to KO$ is the A-hat genus, that induced by $M Spin^c \to KU$ is the Todd genus.
Of course, $KU$ is also a complex oriented cohomology theory and as such canonically comes with a homomorphism of homotopy-commutative ring spectra $MU \to KU$ (this general Prop., see at spin^c structure – from almost complex structure).
On this level there is also the real orientation of real K-theory given by a map
from MR-theory (Kriz 01, (3.20)).
There are multiplicative natural transformations of multiplicative cohomology theories, hence homomorphisms of homotopy-commutative ring spectra
and
which are compatible in that the diagram formed by the evident vertical morphisms
This Conner-Floyd orientation is originally due to Conner-Floyd 66, Section 5.
The morphism MU$\to$KU reflects the complex orientation of KU via the universal complex orientation on MU.
In the context of KK-theory, a morphism $f \colon A \to B$ of C*-algebra, a K-orientation of this morphism is a map
such that (…). The corresponding fiber integration/Gysin map on operator K-theory is then the postcomposition operation
(BMRS 07)
The Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$ induces an equivalence
and $M Spin \to KO$ similarly induces
This is due to (Hopkins-Hovey 92), a variation of the Conner-Floyd isomorphism. See at cobordism theory determining homology theory for more.
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
The MSpin/MSpin^{c}-orientation of KO/KU topological K-theory is attributed to
The MSU/MU-orientation of KO/KU (Conner-Floyd orientation) is originally due to
That ABS-orientation $M Spin^c\to KU$ extendes to a homomorphism of E-infinity rings is due to
Discussion for real K-theory includes
Adams-Novikov spectral sequence_, Topology 40 (2001) 317-399 (pdf)
The discussion of cobordism theory determining homology theory for the K-orientation is due to
Discussion in noncommutative topology/KK-theory is in
no. 6, 1139–1183 (1984)
Last revised on September 22, 2021 at 15:31:57. See the history of this page for a list of all contributions to it.