Contents

cohomology

# Contents

## Idea

A K-orientation is an orientation in generalized cohomology for K-theory (typically: topological K-theory).

## Definition

### In topological K-theory

#### K-Orientation of oriented manifolds

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

$f_! \colon K^\bullet(X) \to K^{\bullet + dim(Y) - dim(X)}(Y) \,.$

#### The universal Atiyah-Bott-Shapiro orientation

There is a universal orientation in generalized cohomology of

hence E-infinity ring homomorphisms out of the Thom spectrum MSpin

MSpin$\longrightarrow$ KO

and

MSpinc$\longrightarrow$ KU

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 an $E_\infty$-map $MU \to KU$ (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

$M \mathbb{R} \longrightarrow K \mathbb{R}$

from MR-theory (Kriz 01, (3.20)).

#### The universal Conner-Floyd orientation

There are multiplicative natural transformations of multiplicative cohomology theories, hence homomorphisms of homotopy-commutative ring spectra

MU$\longrightarrow$ KU

and

MSU$\longrightarrow$ KO

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 KK-theory

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

$f! \in KK_d(B,A)$

such that (…). The corresponding fiber integration/Gysin map on operator K-theory is then the postcomposition operation

$f_! \colon K_\bullet(B) \simeq KK_\bullet(\mathbb{C}, B) \stackrel{f! \circ (-)}{\to} KK_\bullet(\mathbb{C}, A) \simeq K_{\bullet + d}(A) \,.$

(BMRS 07)

## Properties

### As the relation between cobordisms cohomology and K-theory

The Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$ induces an equivalence

$M Spin^c_\bullet(-)\otimes_{M Spin^c_\bullet} KU_\bullet \simeq KU_\bullet(-)$

and $M Spin \to KO$ similarly induces

$M Spin_\bullet(-)\otimes_{M Spin_\bullet} KO_\bullet \simeq KO_\bullet(-)$

This is due to (Hopkins-Hovey 92), a variation of the Conner-Floyd isomorphism. See at cobordism theory determining homology theory for more.

$d$partition function in $d$-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin \to KO$
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

## References

### In topological K-theory

The MSpin/MSpinc-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

• Michael Joachim, Higher coherences for equivariant K-theory, in Structured ring spectra, volume 315 of London Math. Soc. Lecture Note Ser., pages 87–114. Cambridge Univ. Press, Cambridge, 2004 (pdf)

Discussion for real K-theory includes

• Igor Kriz, Real-oriented homotopy theory and an analogue of the

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

### In KK-theory

Discussion in noncommutative topology/KK-theory is in

Last revised on February 18, 2021 at 11:12:51. See the history of this page for a list of all contributions to it.