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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.

The existence of an EE-orientation is necessary in order to have a notion of fiber integration in EE-cohomology.

Definition

Concretely

Definition

Let EE be a multiplicative cohomology theory and let VXV \to X be a topological vector bundle of rank nn. Then an EE-orientation or EE-Thom class on VV is an element of degree nn

uE˜ n(Th(V)) u \in \tilde E^n(Th(V))

in the reduced EE-cohomology ring of the Thom space of VV, such that for every point xXx \in X its restriction i x *ui_x^* u along

i x:S nTh( n)Th(j x)Th(V) i_x \;\colon\; S^n \simeq Th(\mathbb{R}^n) \overset{Th(j_x)}{\longrightarrow} Th(V)

(for nfib xV\mathbb{R}^n \overset{fib_x}{\hookrightarrow} V the fiber of VV over xx) is a generator, in that it is of the form

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

for

  • ϵE˜ 0(S 0)\epsilon \in \tilde E^0(S^0) a unit in E E^\bullet;

  • γ nE˜ n(S n)\gamma_n \in \tilde E^n(S^n) the image of the multiplicative unit under the suspension isomorphism E˜ 0(S 0)E˜ n(S n)\tilde E^0(S^0) \stackrel{\simeq}{\to}\tilde E^n(S^n).

(e.g. Kochmann 96, def. 4.3.4)

Remark

Recall that a (B,f)-structure \mathcal{B} is a system of Serre fibrations B nf nBO(n)B_n \overset{f_n}{\longrightarrow} B O(n) over the classifying spaces for orthogonal structure equipped with maps

j n:B nB n+1 j_n \;\colon\; B_n \longrightarrow B_{n+1}

covering the canonical inclusions of classifying spaces. For instance for G nO(n)G_n \to O(n) a compatible system of topological group homomorphisms, then the (B,f)(B,f)-structure given by the classifying spaces BG nB G_n (possibly suitably resolved for the maps BG nBO(n)B G_n \to B O(n) to become Serre fibrations) defines G-structure.

Given a (B,f)(B,f)-structure, then there are the pullbacks V n f n *(EO(n)×O(n) n)V^{\mathcal{B}}_n \coloneqq f_n^\ast (E O(n)\underset{O(n)}{\times}\mathbb{R}^n) of the universal vector bundles over BO(n)B O(n), which are the universal vector bundles equipped with (B,f)(B,f)-structure

V n EO(n)×O(n) n (pb) B n f n BO(n). \array{ V^{\mathcal{B}}_n &\longrightarrow& E O(n)\underset{O(n)}{\times} \mathbb{R}^n \\ \downarrow &(pb)& \downarrow \\ B_n & \underset{f_n}{\longrightarrow} & B O(n) } \,.

Finally recall that there are canonical morphisms (prop.)

ϕ n:V n V n+1 \phi_n \;\colon\; \mathbb{R} \oplus V^{\mathcal{B}}_n \longrightarrow V^{\mathcal{B}}_{n+1}
Definition

Let EE be a multiplicative cohomology theory and let \mathcal{B} be a multiplicative (B,f)-structure. Then a universal EE-orientation for vector bundles with \mathcal{B}-structure is an EE-orientation, according to def. 1, for each rank-nn universal vector bundle with \mathcal{B}-structure:

ξ nE˜ n(Th(V n ))n \xi_n \in \tilde E^n(Th(V_n^{\mathcal{B}})) \;\;\;\; \forall n \in \mathbb{N}

such that these are compatible in that

  1. for all nn \in \mathbb{N} then

    ξ n=ϕ n *ξ n+1, \xi_n = \phi_n^\ast \xi_{n+1} \,,

    where

    ξ nE˜ n(Th(V n))E˜ n+1(ΣTh(V n))E˜ n+1(Th(V n)) \xi_n \in \tilde E^n(Th(V_n)) \simeq \tilde E^{n+1}(\Sigma Th(V_n)) \simeq \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))

    (with the first isomorphism is the suspension isomorphism of EE and the second exhibiting the homeomorphism of Thom spaces Th(V)ΣTh(V)Th(\mathbb{R} \oplus V)\simeq \Sigma Th(V) (prop.)) and where

    ϕ n *:E˜ n+1(Th(V n+1))E˜ n+1(Th(V n)) \phi_n^\ast \;\colon\; \tilde E^{n+1}(Th(V_{n+1})) \longrightarrow \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))

    is pullback along the canonical ϕ n:V nV n+1\phi_n \colon \mathbb{R}\oplus V_n \to V_{n+1} (prop.).

  2. for all n 1,n 2n_1, n_2 \in \mathbb{N} then

    ξ n+1ξ n+2=ξ n 1+n 2. \xi_{n+1} \cdot \xi_{n+2} = \xi_{n_1 + n_2} \,.
Proposition

A universal EE-orientation, in the sense of def. 2, for vector bundles with (B,f)-structure \mathcal{B}, is equivalently (the homotopy class of) a homomorphism of ring spectra

ξ:ME \xi \;\colon\; M\mathcal{B} \longrightarrow E

from the universal \mathcal{B}-Thom spectrum to a spectrum which via the Brown representability theorem represents the given generalized (Eilenberg-Steenrod) cohomology theory EE (and which we denote by the same symbol).

Proof

The Thom spectrum MM\mathcal{B} has a standard structure of a CW-spectrum. Let now EE denote a sequential Omega-spectrum representing the multiplicative cohomology theory of the same name. Since, in the standard model structure on topological sequential spectra, CW-spectra are cofibrant (prop.) and Omega-spectra are fibrant (thm.) we may represent all morphisms in the stable homotopy category (def.) by actual morphisms

ξ:ME \xi \;\colon\; M \mathcal{B} \longrightarrow E

of sequential spectra (due to this lemma).

Now by definition (def.) such a homomorphism is precissely a sequence of base-point preserving continuous functions

ξ n:(M) n=Th(V n )E n \xi_n \;\colon\; (M\mathcal{B})_n = Th(V_n^{\mathcal{B}}) \longrightarrow E_n

for nn \in \mathbb{N}, such that they are compatible with the structure maps σ n\sigma_n and equivalently with their (S 1()Maps(S 1,) *)(S^1 \wedge(-)\dashv Maps(S^1,-)_\ast)-adjuncts σ˜ n\tilde \sigma_n, in that these diagrams commute:

S 1Th(V n ) S 1ξ n S 1E n σ n M σ n E Th(V n+1 ) ξ n+1 E n+1Th(V n ) ξ n E n σ˜ n M σ˜ n E Maps(S 1,Th(V n+1 )) Maps(S 1,ξ n+1) * Maps(S 1,E n+1) * \array{ S^1 \wedge Th(V^{\mathcal{B}}_n) &\overset{S^1 \wedge \xi_n}{\longrightarrow}& S^1 \wedge E_n \\ {}^{\mathllap{\sigma^{M\mathcal{B}}_n}}\downarrow && \downarrow^{\mathrlap{\sigma^E_n}} \\ Th(V^{\mathcal{B}}_{n+1}) &\underset{\xi_{n+1}}{\longrightarrow}& E_{n+1} } \;\;\;\;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\;\;\;\; \array{ Th(V^{\mathcal{B}}_n) &\overset{\xi_n}{\longrightarrow}& E_n \\ {}^{\mathllap{\tilde \sigma^{M\mathcal{B}}_n}}\downarrow && \downarrow^{\mathrlap{\tilde \sigma^E_n}} \\ Maps(S^1,Th(V^{\mathcal{B}}_{n+1})) &\underset{Maps(S^1,\xi_{n+1})_\ast}{\longrightarrow}& Maps(S^1, E_{n+1})_{\ast} }

for all nn \in \mathbb{N}.

First of all this means (via the identification given by the Brown representability theorem, see this prop.) that the components ξ n\xi_n are equivalently representatives of elements in the cohomology groups

ξ nE˜ n(Th(V n )) \xi_n \in \tilde E^n(Th(V^{\mathcal{B}}_n))

(which we denote by the same symbol, for brevity).

Now by the definition of universal Thom spectra (def., def.), the structure map σ n M\sigma_n^{M\mathcal{B}} is just the map ϕ n:Th(V n )Th(V n+1 )\phi_n \colon \mathbb{R}\oplus Th(V^{\mathcal{B}}_n)\to Th(V_{n+1}^{\mathcal{B}}) from above.

Moreover, by the Brown representability theorem, the adjunct σ˜ n Eξ n\tilde \sigma_n^E \circ \xi_n (on the right) of σ n ES 1ξ n\sigma^E_n \circ S^1 \wedge \xi_n (on the left) is what represents (again by this prop.) the image of

ξ nE n(Th(V n )) \xi_n \in E^n(Th(V^{\mathcal{B}}_n))

under the suspension isomorphism. Hence the commutativity of the above squares is equivalently the first compatibility condition from def. 2: ξ nϕ n *ξ n+1\xi_n \simeq \phi_n^\ast \xi_{n+1} in E˜ n+1(Th(V n ))\tilde E^{n+1}(Th(\mathbb{R}\oplus V_n^{\mathcal{B}}))

Next, ξ\xi being a homomorphism of ring spectra means equivalently (we should be modelling MM\mathcal{B} and EE as structured spectra (here.) to be more precise on this point, but the conclusion is the same) that for all n 1,n 2n_1, n_2\in \mathbb{N} then

Th(V n 1 )Th(T n+2 ) Th(V n 1+n 2) ξ n 1ξ n 2 ξ n 1+n 2 E n 1E n 2 E n 1+n 2. \array{ Th(V_{n_1}^{\mathcal{B}}) \wedge Th(T_{n+2}^{\mathcal{B}}) &\overset{}{\longrightarrow}& Th(V_{n_1 + n_2}) \\ {}^{\mathllap{\xi_{n_1} \wedge \xi_{n_2}}}\downarrow && \downarrow^{\mathrlap{\xi_{n_1 + n_2}}} \\ E_{n_1} \wedge E_{n_2} &\underset{\cdot}{\longrightarrow}& E_{n_1 + n_2} } \,.

This is equivalently the condition ξ n 1ξ n 2ξ n 1+n 2\xi_{n_1} \cdot \xi_{n_2} \simeq \xi_{n_1 + n_2}.

Finally, since MM\mathcal{B} is a ring spectrum, there is an essentially unique multiplicative homomorphism from the sphere spectrum

𝕊eM. \mathbb{S} \overset{e}{\longrightarrow} M\mathcal{B} \,.

This is given by the component maps

e n:S nTh( n)Th(V n ) e_n \;\colon\; S^n \simeq Th(\mathbb{R}^n) \longrightarrow Th(V_{n}^{\mathcal{B}})

that are induced by inclusing the fiber of V n V_{n}^{\mathcal{B}}.

(Maybe we should assume here that \mathcal{B} is G-structure with B n=BG nB_n = B G_n connected and canonically and compatibly pointed.)

Accordingly the composite

𝕊eMξE \mathbb{S} \overset{e}{\longrightarrow} M\mathcal{B} \overset{\xi}{\longrightarrow} E

has as components the restrictions i *ξ ni^\ast \xi_n appearing in def. 1. At the same time, also EE is a ring spectrum, hence it also has an essentially unique multiplicative morphism 𝕊E\mathbb{S} \to E, which hence must agree with i *ξi^\ast \xi, up to homotopy. If we represent EE as a symmetric ring spectrum, then the canonical such has the required property: e 0e_0 is the identity element in degree 0 (being a unit of an ordinary ring, by definition) and hence e ne_n is necessarily its image under the suspension isomorphism, due to compatibility with the structure maps and using the above analysis.

Abstractly

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 we 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 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.

Example

The construction of the string orientation of tmf, hence a morphism

MStringtmf M String \to tmf

is discussed in (Hopkins, last pages).

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.

Relation to cubical structures

For EE a multiplicative weakly periodic 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.

Examples

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
3w4-orientation of EO(2)-theory

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.

References

Discussion in terms of Thom classes:

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

Lecture notes on this 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 27, 2016 14:07:23 by Urs Schreiber (131.220.184.222)