# nLab orientation in generalized cohomology

cohomology

### Theorems

#### Integration theory

integration

analytic integrationcohomological integration
measureorientation in generalized cohomology
Riemann/Lebesgue integration, of differential formspush-forward in generalized cohomology/in differential cohomology

# 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 $P=\mathrm{Th}\left(V\right)$ 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 $𝕊$ for the sphere spectrum.

#### ${\mathrm{GL}}_{1}\left(R\right)$-principal $\infty$-bundles

Write ${R}^{×}$ or ${\mathrm{GL}}_{1}\left(R\right)$ 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 ${\pi }_{0}\left(R\right)$.

Since $R$ is ${E}_{\infty }$, the space ${\mathrm{GL}}_{1}\left(R\right)$ is itself an infinite loop space. Its one-fold delooping $B{\mathrm{GL}}_{1}\left(R\right)$ is the classifying space for ${\mathrm{GL}}_{1}\left(R\right)$-principal ∞-bundles (in Top): for $X\in \mathrm{Top}$ and $\zeta :X\to B{\mathrm{GL}}_{1}\left(R\right)$ a map, its homotopy fiber

$\begin{array}{ccccc}{\mathrm{GL}}_{1}\left(R\right)& \to & P& \to & *\\ ↓& & ↓& & ↓\\ *& \stackrel{x}{\to }& X& \stackrel{\zeta }{\to }& B{\mathrm{GL}}_{1}\left(R\right)\end{array}$\array{ GL_1(R) &\to& P &\to& * \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{x}{\to}& X &\stackrel{\zeta}{\to}& B GL_1(R) }

is the ${\mathrm{GL}}_{1}\left(R\right)$-principal $\infty$-bundle $P\to X$ classified by that map.

###### Example

For $R=𝕊$ the sphere spectrum, we have that $B{\mathrm{GL}}_{1}\left(𝕊\right)$ is the classifying space for spherical fibrations.

###### Example

There is a canonical morphism

$BO\to B{\mathrm{GL}}_{1}\left(𝕊\right)$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

$J:V↦{S}^{V}$J : V \mapsto S^V

which sends each fiber to its one-point compactification.

#### ${\mathrm{GL}}_{1}\left(R\right)$-associated $\infty$-bundles

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

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

the associated bundle is the smash product over ${\Sigma }^{\infty }{\mathrm{GL}}_{1}\left(R\right)$

${X}^{\zeta }:={\Sigma }^{\infty }P{\wedge }_{{\Sigma }^{\infty }{\mathrm{GL}}_{1}\left(R\right)}R\phantom{\rule{thinmathspace}{0ex}}.$X^\zeta := \Sigma^\infty P \wedge_{\Sigma^\infty GL_1(R)} R \,.

This is the generalized Thom spectrum. For $R=KO$ 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 $\zeta :X\to B{\mathrm{GL}}_{1}\left(R\right)$ the classifying map for an $R$-bundle, the corresponding associated $S$-bundle classified by the composite

$X\stackrel{\zeta }{\to }B{\mathrm{GL}}_{1}\left(R\right)\stackrel{f}{\to }B{\mathrm{GL}}_{1}\left(S\right)$X \stackrel{\zeta}{ \to } B GL_1(R) \stackrel{f}{\to} B GL_1(S)

is given by the smash product

${X}^{f\circ \zeta }\simeq {X}^{\zeta }{\wedge }_{R}S\phantom{\rule{thinmathspace}{0ex}}.$X^{f \circ \zeta} \simeq X^\zeta \wedge_R S \,.

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

#### $R$-Orientations

For $X\stackrel{\zeta }{\to }B{\mathrm{GL}}_{1}\left(𝕊\right)$ 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

$X\stackrel{\zeta }{\to }B{\mathrm{GL}}_{1}\left(𝕊\right)\stackrel{\iota }{\to }B{\mathrm{GL}}_{1}\left(R\right)\phantom{\rule{thinmathspace}{0ex}},$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 $𝕊\to R$ (the sphere spectrum is the initial object in ${E}_{\infty }$-rings).

Specifically, for $V:X\to BO$ 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

$\begin{array}{ccccc}BO& \stackrel{J}{\to }& B{\mathrm{GL}}_{1}\left(𝕊\right)& \stackrel{\iota }{\to }& B{\mathrm{GL}}_{1}\left(R\right)\\ {}^{V}↑& {↗}_{\zeta }\\ X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$BO\stackrel{J}{\to }B{\mathrm{GL}}_{1}\left(𝕊\right)\stackrel{\iota }{\to }B{\mathrm{GL}}_{1}\left(R\right)\phantom{\rule{thinmathspace}{0ex}}.$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

$B\mathrm{Spin}\to BO\stackrel{J}{\to }B{\mathrm{GL}}_{1}\left(𝕊\right)\stackrel{\iota }{\to }B{\mathrm{GL}}_{1}\left(R\right)$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

$B\mathrm{String}\to B\mathrm{Spin}\to BO\stackrel{J}{\to }B{\mathrm{GL}}_{1}\left(𝕊\right)\stackrel{\iota }{\to }B{\mathrm{GL}}_{1}\left(R\right)$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 $BO$.

Now, the Thom spectrum $MO$ is the sphere bundle over $BO$ 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

$BO\stackrel{J}{\to }B{\mathrm{GL}}_{1}\left(𝕊\right)\stackrel{\iota }{\to }B{\mathrm{GL}}_{1}\left(R\right)$B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)

correspond to ${E}_{\infty }$-maps

$MO\to R$M O \to R

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

$B\mathrm{Spin}\to BO\stackrel{J}{\to }B{\mathrm{GL}}_{1}\left(𝕊\right)\stackrel{\iota }{\to }B{\mathrm{GL}}_{1}\left(R\right)$B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)

corresponds to morphisms

$M\mathrm{Spin}\to R$M Spin \to R

and trivializations of

$B\mathrm{String}\to B\mathrm{Spin}\to BO\stackrel{J}{\to }B{\mathrm{GL}}_{1}\left(𝕊\right)\stackrel{\iota }{\to }B{\mathrm{GL}}_{1}\left(R\right)$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

$M\mathrm{String}\to R$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

$M\mathrm{String}\to \mathrm{tmf}$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 $u\in {H}^{n}\left(\mathrm{Th}\left(V\right)\right)$ in the cohomology of the Thom space of $V$ – a Thom class – with the property that its restriction ${i}^{*}u$ along $i:{S}^{n}\to \mathrm{Th}\left(V\right)$ to any fiber of $\mathrm{Th}\left(V\right)$ is

${i}^{*}u=ϵ\cdot {\gamma }_{n}\phantom{\rule{thinmathspace}{0ex}},$i^* u = \epsilon \cdot \gamma_n \,,

where

• $ϵ\in {H}^{0}\left({S}^{0}\right)$ is a multiplicatively invertible element;

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

Multiplication with $u$ induces hence an isomorphism

$\left(-\right)\cdot u:{H}^{•}\left(X\right)\stackrel{\simeq }{\to }{H}^{•+n}\left(\mathrm{Th}\left(V\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$(-)\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 $V\wedge H\left(ℝ\right)$ of smashing with the Eilenberg-MacLane spectrum.

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

• K-orientation

• string orientation of tmf

• 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 18, 2013 18:49:35 by Urs Schreiber (131.174.43.49)