# nLab orientation in generalized cohomology

cohomology

### Theorems

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

integration

# Contents

## Idea

Generally, for $E$ an E-∞ ring spectrum, and $P \to X$ a sphere spectrum-bundle, an $E$-orientation of $P$ is a trivialization of the associated $E$-bundle.

Specifically, for $P = Th(V)$ the Thom space of a vector bundle $V \to X$, an $E$-orientation of $V$ is an $E$-orientation of $P$.

More generally, for $A$ an $E$-algebra spectrum, an $E$-bundle is $A$-orientable if the associated $A$-bundle is trivializable. For more on this see (∞,1)-vector bundle.

## Definition

### General abstract

Let $E$ be a E-∞ ring spectrum. Write $\mathbb{S}$ for the sphere spectrum.

#### $GL_1(R)$-principal $\infty$-bundles

Write $R^\times$ or $GL_1(R)$ for the general linear group of the $E_\infty$-ring $R$: it is the subspace of the degree-0 space $\Omega^\infty R$ on those points that map to multiplicatively invertible elements in the ordinary ring $\pi_0(R)$.

Since $R$ is $E_\infty$, the space $GL_1(R)$ is itself an infinite loop space. Its one-fold delooping $B GL_1(R)$ is the classifying space for $GL_1(R)$-principal ∞-bundles (in Top): for $X \in Top$ and $\zeta : X \to B GL_1(R)$ a map, its homotopy fiber

$\array{ GL_1(R) &\to& P &\to& * \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{x}{\to}& X &\stackrel{\zeta}{\to}& B GL_1(R) }$

is the $GL_1(R)$-principal $\infty$-bundle $P \to X$ classified by that map.

###### Example

For $R = \mathbb{S}$ the sphere spectrum, we have that $B GL_1(\mathbb{S})$ is the classifying space for spherical fibrations.

###### Example

There is a canonical morphism

$B O \to B GL_1(\mathbb{S})$

from the classifying space of the orthogonal group to that of the infinity-group of units of the sphere spectrum, called the J-homomorphism. Postcomposition with this sends real vector bundles $V \to X$ to sphere bundles. This is what is modeled by the Thom space construction

$J : V \mapsto S^V$

which sends each fiber to its one-point compactification.

#### $GL_1(R)$-associated $\infty$-bundles

For $P \to X$ a $GL_1(R)$-principal ∞-bundle there is canonically the corresponding associated ∞-bundle with fiber $R$. Precisely, in the stable (∞,1)-category $Stab(Top)$ of spectra, regarded as the stabilization of the (∞,1)-topos Top

$Stab(Top) \simeq Spectra \stackrel{\overset{\Sigma^\infty}{\leftarrow}}{\underset{\Omega^\infty}{\to}} Top$

the associated bundle is the smash product over $\Sigma^\infty GL_1(R)$

$X^\zeta := \Sigma^\infty P \wedge_{\Sigma^\infty GL_1(R)} R \,.$

This is the generalized Thom spectrum. For $R = K O$ the real K-theory spectrum this is given by the ordinary Thom space construction on a vector bundle $V \to X$.

An $E$-orientation of a vector bundle $V \to X$ is a trivialization of the $E$-module bundle $E \wedge S^V$, where fiberwise form the smash product of $E$ with the Thom space of $V$.

###### Proposition

For $f : R \to S$ a morphism of $E_\infty$-rings, and $\zeta : X \to B GL_1(R)$ the classifying map for an $R$-bundle, the corresponding associated $S$-bundle classified by the composite

$X \stackrel{\zeta}{ \to } B GL_1(R) \stackrel{f}{\to} B GL_1(S)$

is given by the smash product

$X^{f \circ \zeta} \simeq X^\zeta \wedge_R S \,.$

This appears as (Hopkins, bottom of p. 6).

#### $R$-Orientations

For $X \stackrel{\zeta}{\to} B GL_1(\mathbb{S})$ a sphere bundle, an $E$-orientation on $X^\zeta$ is a trivialization of the associated $R$-bundle $X^\zeta \wedge R$, hence a trivialization (null-homotopy) of the classifying morphism

$X \stackrel{\zeta}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R) \,,$

where the second map comes from the unit of $E_\infty$-rings $\mathbb{S} \to R$ (the sphere spectrum is the initial object in $E_\infty$-rings).

Specifically, for $V : X \to B O$ a vector bundle, an $E$-orientation on it is a trivialization of the $R$-bundle associated to the associated Thom space sphere bundle, hence a trivialization of the morphism

$\array{ B O &\stackrel{J}{\to}& B GL_1(\mathbb{S}) &\stackrel{\iota}{\to}& B GL_1(R) \\ {}^{\mathllap{V}}\uparrow & \nearrow_{\mathrlap{\zeta}} \\ X } \,.$

This appears as (Hopkins, p.7).

A natural $R$-orientation of all vector bundles is therefore a trivialization of the morphism

$B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R) \,.$

Similarly, an $R$-orientation of all spinor bundles is a trivialization of

$B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)$

and an $R$-orientation of all string group-bundles a trivialization of

$B String \to B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)$

and so forth, through the Whitehead tower of $B O$.

Now, the Thom spectrum $M O$ is the sphere bundle over $B O$ associated to the $O$-universal principal bundle. In generalization of the way that a trivialization of an ordinary $G$-principal bundle $P$ is given by a $G$-equivariant map $P \to G$, one finds that trivializations of the morphism

$B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)$

correspond to $E_\infty$-maps

$M O \to R$

from the Thom spectrum to $R$. Similarly trivialization of

$B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)$

corresponds to morphisms

$M Spin \to R$

and trivializations of

$B String \to B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)$

to morphisms

$M String \to R$

and so forth.

This is the way orientations in generalized cohomology often appear in the literature.

###### Example

The construction of the string orientation of tmf, hence a morphism

$M String \to tmf$

is discussed in (Hopkins, last pages).

### Concretely for vector bundles

For $H$ the multiplicative cohomology theory corresponding to $E$, and $V \to X$ a vector bundle of rank $n$, an $H$-orientation of $V$ is an element $u \in H^n(Th(V))$ in the cohomology of the Thom space of $V$ – a Thom class – with the property that its restriction $i^* u$ along $i : S^n \to Th(V)$ to any fiber of $Th(V)$ is

$i^* u = \epsilon \cdot \gamma_n \,,$

where

• $\epsilon \in H^0(S^0)$ is a multiplicatively invertible element;

• $\gamma_n \in H^n(S^n)$ is the image of the multiplicative unit under the suspension isomorphism $H^0(S^0) \stackrel{\simeq}{\to}H^n(S^n)$.

Multiplication with $u$ induces hence an isomorphism

$(-)\cdot u : H^\bullet(X) \stackrel{\simeq}{\to} H^{\bullet + n}(Th(V)) \,.$

This is called the Thom isomorphism.

The existence of an $H$-orientation is necessary in order to have a notion of fiber integration in $H$-cohomology.

## Examples

• For $V \to X$ a vector bundle of rank $k$, an ordinary orientation is a trivialization of the line bundle $\wedge^k V$. This is indeed equivalently a trivialization of $V \wedge H(\mathbb{R})$ of smashing with the Eilenberg-MacLane spectrum.

• For spin orientation of vector bundle the construction is given by forming Clifford algebra bundles.

• K-orientation

• For string orientation of vector bundle the construction is supposed to be given by forming free fermion local net-bundle. See Andre Henriques’ website.

(…)

## References

A comprehesive account is in

The general abstract story of $E$-orientation of sphere fibrations is discussed in

with an eye towards constructing the string structure-orientation of tmf.

Orientation of vector bundles in $E$-cohomology is discussed for instance in

Revised on June 27, 2013 10:21:28 by Urs Schreiber (80.90.61.2)