cohomology

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 $R$-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 MO is the spherical fibration 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.

## Properties

### Relation between Thom classes and trivializations

The relation (equivalence) between choices of Thom classes and trivializations of (∞,1)-line bundles is discussed e.g. in Ando-Hopkins-Rezk 10, section 3.3

### Relation to genera

Let $G$ be a topological group equipped with a homomorphism to the stable orthogonal group, and write $B G \to B O$ for the corresponding map of classifying spaces. Write $J \colon B O \longrightarrow B GL_1(\mathbb{S})$ for the J-homomorphism.

For $E$ an E-∞ ring, there is a canonical homomorphism $B GL_1(\mathbb{S}) \to B GL_1(E)$ between the deloopings of the ∞-groups of units. A trivialization of the total composite

$B G \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to B GL_1(E)$

is a universal $E$-orientation of G-structures. Under (∞,1)-colimit in $E Mod$ this induces a homomorphism of $E$-∞-modules

$\sigma \;\colon\; M G \to E$

from the universal $G$-Thom spectrum to $E$.

If here $G \to GL_1(\mathbb{S})$ is the $\Omega^\infty$-component of a map of spectra then this $\sigma$ is a homomorphism of E-∞ rings and in this case there is a bijection between universal orientations and such $E_\infty$-ring homomorphisms (Ando-Hopkins-Rezk 10, prop. 2.11).

The latter, on passing to homotopy groups, are genera on manifolds with G-structure.

### Relation to cubical structures

For $E$ a multiplicative weakly perioduc complex orientable cohomology theory then $Spec E^0(B U\langle 6\rangle)$ is naturally equivalent to the space of cubical structures on the trivial line bundle over the formal group of $E$.

In particular, homotopy classes of maps of E-infinity ring spectra $MU\angle 6\rangle \to E$ from the Thom spectrum to $E$, and hence universal $MU\langle 6\rangle$-orientations (see there) of $E$ are in natural bijection with these cubical structures.

See at cubical structure for more details and references. This way for instance the string orientation of tmf has been constructed. See there for more.

## Examples

$d$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

### Complex orientation

An $E_\infty$ complex oriented cohomology theory $E$ is indeed equipped with a universal complex orientation given by an $E_\infty$-ring homomorphism $MU \to E$, see here.

## References

A comprehensive account of the general abstract persepctive (with predecessors in Ando-Hopkins-Rezk 10) is in

Lecture notes include

which are motivated towards constructing the string orientation of tmf, based on

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

Revised on May 15, 2014 17:22:41 by Jon Beardsley (68.49.92.24)