orientation in generalized cohomology




Special and general types

Special notions


Extra structure



Index theory

Integration theory



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.


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.


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.


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.


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).


For XζBGL 1(𝕊)X \stackrel{\zeta}{\to} B GL_1(\mathbb{S}) a sphere bundle, an RR-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.


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 \,,


  • 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.


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.

Relation to cubical structures

For EE a multiplicative weakly perioduc complex orientable cohomology theory then SpecE 0(BU6)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 EE.

In particular, homotopy classes of maps of E-infinity ring spectra MU6EMU\angle 6\rangle \to E from the Thom spectrum to EE, and hence universal MU6MU\langle 6\rangle-orientations (see there) of EE 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.


partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

ddpartition function in dd-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 MSpinKOM 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 MSpin cKUM 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 MSpin cKUM 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 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.


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 May 15, 2014 17:22:41 by Jon Beardsley (