# nLab orientation in generalized cohomology

Contents

cohomology

### Theorems

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

# 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 $P = Th(V)$ 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.

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

## Definition

### Concretely

#### Orientation of a vector bundle

###### Definition

Let $E$ be a multiplicative cohomology theory and let $V \to X$ be a topological vector bundle of rank $n$. Then an $E$-orientation or $E$-Thom class on $V$ is an element of degree $n$

$u \in \tilde E^n(Th(V))$

in the reduced $E$-cohomology ring of the Thom space of $V$, such that for every point $x \in X$ its restriction $i_x^* u$ along

$i_x \;\colon\; S^n \simeq Th(\mathbb{R}^n) \overset{Th(j_x)}{\longrightarrow} Th(V)$

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

$i^\ast u = \epsilon \cdot \gamma_n$

for

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

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

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

#### Orientation of a manifold

###### Definition

Let $E$ be a multiplicative cohomology theory and let $X$ be a manifold, possibly with boundary, of dimension $n$. An $E$-orientation of $X$ is a class in the $E$-generalized homology

$\iota \in E_n(X,\partial X)$

with the property that for each point $x \in Int(X)$ in the interior, it maps to a generator of $E_\bullet(\ast)$ under the map

$E_\bullet(X,\partial X) \longrightarrow E_\bullet(X,\; X - \{x\}) \simeq E_\bullet(D^n, S^{n-1}) \simeq E_{\bullet-n}(\ast) \,,$

where the isomorphism is the excision isomorphism (def.) for the complement of a closed n-ball around $x$.

(e.g. Kochman 96, p. 134)

###### Proposition

$E$-orientations of manifolds (def. ) are equivalent to $E$-orientations of their stable normal bundle (def. ).

(e.g. Rudyak 98, chapter V, theorem 2.4) (also Kochman 96, prop. 4.3.5, but maybe that proof needs an extra argument)

#### Universal orientation of vector bundles

###### Remark

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

$j_n \;\colon\; B_n \longrightarrow B_{n+1}$

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

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

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

Finally recall that there are canonical morphisms (prop.)

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

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

$\xi_n \in \tilde E^n(Th(V_n^{\mathcal{B}})) \;\;\;\; \forall n \in \mathbb{N}$

such that these are compatible in that

1. for all $n \in \mathbb{N}$ then

$\xi_n = \phi_n^\ast \xi_{n+1} \,,$

where

$\xi_n \in \tilde E^n(Th(V_n)) \simeq \tilde E^{n+1}(\Sigma Th(V_n)) \simeq \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))$

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

$\phi_n^\ast \;\colon\; \tilde E^{n+1}(Th(V_{n+1})) \longrightarrow \tilde E^{n+1}(Th(\mathbb{R}\oplus V_n))$

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

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

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

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

$\xi \;\colon\; M\mathcal{B} \longrightarrow E$

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

###### Proof

The Thom spectrum $M\mathcal{B}$ has a standard structure of a CW-spectrum. Let now $E$ 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

$\xi \;\colon\; M \mathcal{B} \longrightarrow E$

of sequential spectra (due to this lemma).

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

$\xi_n \;\colon\; (M\mathcal{B})_n = Th(V_n^{\mathcal{B}}) \longrightarrow E_n$

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

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

for all $n \in \mathbb{N}$.

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

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

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

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

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

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

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

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

$\array{ Th(V_{n_1}^{\mathcal{B}}) \wedge Th(V_{n_2}^{\mathcal{B}}) &\overset{}{\longrightarrow}& Th(V_{n_1 + n_2}) \\ {}^{\mathllap{\xi_{n_1} \wedge \xi_{n_2}}}\downarrow && \downarrow^{\mathrlap{\xi_{n_1 + n_2}}} \\ E_{n_1} \wedge E_{n_2} &\underset{\cdot}{\longrightarrow}& E_{n_1 + n_2} } \,.$

This is equivalently the condition $\xi_{n_1} \cdot \xi_{n_2} \simeq \xi_{n_1 + n_2}$.

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

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

This is given by the component maps

$e_n \;\colon\; S^n \simeq Th(\mathbb{R}^n) \longrightarrow Th(V_{n}^{\mathcal{B}})$

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

Accordingly the composite

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

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

### Abstractly

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

#### $GL_1(R)$-principal $\infty$-bundles

Write $R^\times$ or $GL_1(R)$ 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 $\pi_0(R)$.

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

$\array{ GL_1(R) &\to& P &\to& * \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{x}{\to}& X &\stackrel{\zeta}{\to}& B GL_1(R) }$

is the $GL_1(R)$-principal $\infty$-bundle $P \to X$ classified by that map.

###### Example

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

###### Example

There is a canonical morphism

$B O \to B GL_1(\mathbb{S})$

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

$J : V \mapsto S^V$

which sends each fiber to its one-point compactification.

#### $GL_1(R)$-associated $\infty$-bundles

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

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

the associated bundle is the smash product over $\Sigma^\infty GL_1(R)$

$X^\zeta := \Sigma^\infty P \wedge_{\Sigma^\infty GL_1(R)} R \,.$

This is the generalized Thom spectrum. For $R = K O$ 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 we 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 $\zeta : X \to B GL_1(R)$ the classifying map for an $R$-bundle, the corresponding associated $S$-bundle classified by the composite

$X \stackrel{\zeta}{ \to } B GL_1(R) \stackrel{f}{\to} B GL_1(S)$

is given by the smash product

$X^{f \circ \zeta} \simeq X^\zeta \wedge_R S \,.$

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

#### $R$-Orientations

For $X \stackrel{\zeta}{\to} B GL_1(\mathbb{S})$ a sphere bundle, an $R$-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

$X \stackrel{\zeta}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R) \,,$

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

Specifically, for $V : X \to B O$ 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

$\array{ B O &\stackrel{J}{\to}& B GL_1(\mathbb{S}) &\stackrel{\iota}{\to}& B GL_1(R) \\ {}^{\mathllap{V}}\uparrow & \nearrow_{\mathrlap{\zeta}} \\ X } \,.$

This appears as (Hopkins, p.7).

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

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

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

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

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

$B String \to B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)$

and so forth, through the Whitehead tower of $B O$.

Now, the Thom spectrum MO is the spherical fibration over $B O$ 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

$B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)$

correspond to $E_\infty$-maps

$M O \to R$

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

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

corresponds to morphisms

$M Spin \to R$

and trivializations of

$B String \to B Spin \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \stackrel{\iota}{\to} B GL_1(R)$

to morphisms

$M String \to R$

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

$M String \to tmf$

is discussed in (Hopkins, last pages).

## Properties

### Relation to top Chern class

###### Proposition

The top Chern class of a complex vector bundle $\mathcal{V}_X$ equals the pullback of any Thom class $th \;\in\; H^{2n}\big( \mathcal{V}_X; \mathbb{Z} \big)$ on $\mathcal{V}_X$ along the zero-section:

$\mathcal{V}_X \; \text{has complex rank}\;n \;\;\;\;\; \Rightarrow \;\;\;\;\; c_n \big( \mathcal{V}_X \big) \;\;\; = \;\;\; (0_X)^\ast (th) \;\; \in \; H^{2n} \big( X ; \, \mathbb{Z} \big)$

### Relation to the Todd class

###### Proposition

(rational Todd class is Chern character of Thom class)

Let $V \to X$ be a complex vector bundle over a compact topological space. Then the Todd class $Td(V) \,\in\, H^{ev}(X; \mathbb{Q})$ of $V$ in rational cohomology equals the Chern character $ch$ of the Thom class $th(V) \,\in\, K\big( Th(V) \big)$ in the complex topological K-theory of the Thom space $Th(V)$, when both are compared via the Thom isomorphisms $\phi_E \;\colon\; E(X) \overset{\simeq}{\to} E\big( Th(V)\big)$:

$\phi_{H\mathbb{Q}} \big( Td(V) \big) \;=\; ch\big( th(V) \big) \,.$

More generally , for $x \in K(X)$ any class, we have

$\phi_{H\mathbb{Q}} \big( ch(x) \cup Td(V) \big) \;=\; ch\big( \phi_{K}(x) \big) \,,$

which specializes to the previous statement for $x = 1$.

### Relation to 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 $G$ be a topological group equipped with a homomorphism to the stable orthogonal group, and write $B G \to B O$ for the corresponding map of classifying spaces. Write $J \colon B O \longrightarrow B GL_1(\mathbb{S})$ for the J-homomorphism.

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

$B G \to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to B GL_1(E)$

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

$\sigma \;\colon\; M G \to E$

from the universal $G$-Thom spectrum to $E$.

If here $G \to GL_1(\mathbb{S})$ is the $\Omega^\infty$-component of a map of spectra then this $\sigma$ is a homomorphism of E-∞ rings and in this case there is a bijection between universal orientations and such $E_\infty$-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 $E$ a multiplicative weakly periodic complex orientable cohomology theory then $Spec E^0(B U\langle 6\rangle)$ is naturally equivalent to the space of cubical structures on the trivial line bundle over the formal group of $E$.

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

$d$partition function in $d$-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 $M Spin \to KO$
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 $M Spin^c \to KU$
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
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 $E_\infty$ complex oriented cohomology theory $E$ is indeed equipped with a universal complex orientation given by an $E_\infty$-ring homomorphism $MU \to E$, see here.

Discussion in terms of Thom classes:

A comprehensive account of the general abstract perspective (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

For an informal exposition in terms of spectra, see

• Mattia Coloma, Domenico Fiorenza, Eugenio Landi, An exposition of the topological half of the Grothendieck-Hirzebruch-Riemann-Roch theorem in the fancy language of spectra, (arXiv:1911.12035)