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

${\mathrm{GL}}_1(R)$-principal $\mathrm{\infty }$-bundles

Write $R^{\times }$ or ${\mathrm{GL}}_1(R)$ for the general linear group of the $E_{\mathrm{\infty }}$-ring $R$: it is the subspace of the degree-0 space ${\Omega }^{\mathrm{\infty }}R$ on those points that map to multiplicatively invertible elements in the ordinary ring ${\pi }_0(R)$.

Since $R$ is $E_{\mathrm{\infty }}$, the space ${\mathrm{GL}}_1(R)$ is itself an infinite loop space. Its one-fold delooping $B{\mathrm{GL}}_1(R)$ is the classifying space for ${\mathrm{GL}}_1(R)$-principal ∞-bundles (in Top): for $X\in \mathrm{Top}$ and $\zeta :X\to B{\mathrm{GL}}_1(R)$ a map, its homotopy fiber

is the ${\mathrm{GL}}_1(R)$-principal $\mathrm{\infty }$-bundle $P\to X$ classified by that map.

${\mathrm{GL}}_1(R)$-associated $\mathrm{\infty }$-bundles

For $P\to X$ a ${\mathrm{GL}}_1(R)$-principal ∞-bundle there is canonically the corresponding associated ∞-bundle with fiber $R$. Precisely, in the stable (∞,1)-category $\mathrm{Stab}(\mathrm{Top})$ of spectra, regarded as the stabilization of the (∞,1)-topos Top

the associated bundle is the smash product over ${\Sigma }^{\mathrm{\infty }}{\mathrm{GL}}_1(R)$

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 $V$.

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

$R$-Orientations

For $X\stackrel{\zeta }{\to }B{\mathrm{GL}}_1(𝕊)$ 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

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

This appears as (Hopkins, p.7).

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

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

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

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

correspond to $E_{\mathrm{\infty }}$-maps

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

corresponds to morphisms

and trivializations of

to morphisms

and so forth.

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

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

where

• $ϵ\in H^0(S^0)$ 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

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 $V\wedge H(ℝ)$ of smashing with the Eilenberg-MacLane spectrum.

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

• K-orientation

• 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.

(…)

Related concepts

• differential Thom class

• differential orientation, fiber integration in differential cohomology

• (infinity,1)-vector bundle

References

A comprehesive account is in

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)

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

• Mike Hopkins (notes byAndre Henriques), The String orientation of tmf ) (arXiv:0805.0743)

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

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

• eom, Orientation