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orientation in generalized cohomology

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Idea

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

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

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

Definition

General abstract

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

GL 1(R)GL_1(R)-principal \infty-bundles

Write R ×R^\times or GL 1(R)GL_1(R) for the general linear group of the E E_\infty-ring RR: it is the subspace of the degree-0 space Ω R\Omega^\infty R on those points that map to multiplicatively invertible elements in the ordinary ring π 0(R)\pi_0(R).

Since RR is E E_\infty, the space GL 1(R)GL_1(R) is itself an infinite loop space. Its one-fold delooping BGL 1(R)B GL_1(R) is the classifying space for GL 1(R)GL_1(R)-principal ∞-bundles (in Top): for XTopX \in Top and ζ:XBGL 1(R)\zeta : X \to B GL_1(R) a map, its homotopy fiber

GL 1(R) P * * x X ζ BGL 1(R) \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)GL_1(R)-principal \infty-bundle PXP \to X classified by that map.

Example

For R=𝕊R = \mathbb{S} the sphere spectrum, we have that BGL 1(𝕊)B GL_1(\mathbb{S}) is the classifying space for spherical fibrations.

Example

There is a canonical morphism

BOBGL 1(𝕊) 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 VXV \to X to sphere bundles. This is what is modeled by the Thom space construction

J:VS V J : V \mapsto S^V

which sends each fiber to its one-point compactification.

GL 1(R)GL_1(R)-associated \infty-bundles

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

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

the associated bundle is the smash product over Σ GL 1(R)\Sigma^\infty GL_1(R)

X ζ:=Σ P Σ GL 1(R)R. X^\zeta := \Sigma^\infty P \wedge_{\Sigma^\infty GL_1(R)} R \,.

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

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

Proposition

For f:RSf : R \to S a morphism of E E_\infty-rings, and ζ:XBGL 1(R)\zeta : X \to B GL_1(R) the classifying map for an RR-bundle, the corresponding associated SS-bundle classified by the composite

XζBGL 1(R)fBGL 1(S) X \stackrel{\zeta}{ \to } B GL_1(R) \stackrel{f}{\to} B GL_1(S)

is given by the smash product

X fζX ζ RS. X^{f \circ \zeta} \simeq X^\zeta \wedge_R S \,.

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

RR-Orientations

For XζBGL 1(𝕊)X \stackrel{\zeta}{\to} B GL_1(\mathbb{S}) a sphere bundle, an EE-orientation on X ζX^\zeta is a trivialization of the associated RR-bundle X ζRX^\zeta \wedge R, hence a trivialization (null-homotopy) of the classifying morphism

XζBGL 1(𝕊)ιBGL 1(R), 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 E_\infty-rings 𝕊R\mathbb{S} \to R (the sphere spectrum is the initial object in E E_\infty-rings).

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

BO J BGL 1(𝕊) ι BGL 1(R) V ζ X. \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 RR-orientation of all vector bundles is therefore a trivialization of the morphism

BOJBGL 1(𝕊)ιBGL 1(R). B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R) \,.

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

BSpinBOJBGL 1(𝕊)ιBGL 1(R) B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)

and an RR-orientation of all string group-bundles a trivialization of

BStringBSpinBOJBGL 1(𝕊)ιBGL 1(R) 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 BOB O.

Now, the Thom spectrum MO is the spherical fibration over BOB O associated to the OO-universal principal bundle. In generalization of the way that a trivialization of an ordinary GG-principal bundle PP is given by a GG-equivariant map PGP \to G, one finds that trivializations of the morphism

BOJBGL 1(𝕊)ιBGL 1(R) B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)

correspond to E E_\infty-maps

MOR M O \to R

from the Thom spectrum to RR. Similarly trivialization of

BSpinBOJBGL 1(𝕊)ιBGL 1(R) B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)

corresponds to morphisms

MSpinR M Spin \to R

and trivializations of

BStringBSpinBOJBGL 1(𝕊)ιBGL 1(R) 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

MStringR 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

MStringtmf M String \to tmf

is discussed in (Hopkins, last pages).

Concretely for vector bundles

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

i *u=ϵγ n, i^* u = \epsilon \cdot \gamma_n \,,

where

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

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

Multiplication with uu induces hence an isomorphism

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

This is called the Thom isomorphism.

The existence of an HH-orientation is necessary in order to have a notion of fiber integration in HH-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 GG be a topological group equipped with a homomorphism to the stable orthogonal group, and write BGBOB G \to B O for the corresponding map of classifying spaces. Write J:BOBGL 1(𝕊)J \colon B O \longrightarrow B GL_1(\mathbb{S}) for the J-homomorphism.

For EE an E-∞ ring, there is a canonical homomorphism BGL 1(𝕊)BGL 1(E)B GL_1(\mathbb{S}) \to B GL_1(E) between the deloopings of the ∞-groups of units. A trivialization of the total composite

BGBOJBGL 1(𝕊)BGL 1(E) B G \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to B GL_1(E)

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

σ:MGE \sigma \;\colon\; M G \to E

from the universal GG-Thom spectrum to EE.

If here GGL 1(𝕊)G \to GL_1(\mathbb{S}) is the Ω \Omega^\infty-component of a map of spectra then this σ\sigma is a homomorphism of E-∞ rings and in this case there is a bijection between universal orientations and such E 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.

Examples

Atiyah-Bott-Shapiro orientation

SpinSpin-orientation of KOKO

Spin cSpin^c-orientation of KUKU

Complex orientation

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

Ando-Hopkins-Rezk string orientation of tmftmf

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 EE-cohomology is discussed for instance in

Revised on March 29, 2014 04:29:53 by Urs Schreiber (185.37.147.12)