# nLab Thom spectrum

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

## Theorem

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

The universal real Thom spectrum MO is a connective spectrum whose associated infinite loop space is the classifying space for cobordism:

$\Omega^\infty M O \simeq \vert Cob_\infty \vert .$

In particular, $\pi_n M O$ is naturally identified with the set of cobordism classes of closed $n$-manifolds.

More abstractly, MO is the homotopy colimit of the J-homomorphism in Spectra

$M O \simeq \underset{\longrightarrow}{\lim}(B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to Spectra)$

hence the “total space” of the universal spherical fibration on the classifying space $B O$ for (stable) real vector bundles.

Given this, for any topological group $G$ equipped with a homomorphism to the orthogonal group there is a corresponding Thom spectrum

$M G \simeq \underset{\longrightarrow}{\lim}(B G\to B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to Spectra) \,.$

This is considered particularly for the stages $G$ in the Whitehead tower of the orthogonal group, where it yields $M$Spin, $M$String group, etc.

All these Thom spectra happen to naturally have the structure of E-∞ rings and $E_\infty$-ring homomorphisms $M O\to E$ into another $E_\infty$-ring $E$ are equivalently universal orientations in E-cohomology. On homotopy groups these are genera with coefficients in the underlying ring $\pi_\bullet(E)$.

## Definition

### For vector bundles

For $V \to X$ a vector bundle, we have a weak homotopy equivalence

$Th(\mathbb{R}^n \oplus V) \simeq S^n \wedge Th(V) \simeq \Sigma^n Th(V)$

between, on the one hand, the Thom space of the direct sum of $V$ with the trivial vector bundle of rank $n$ and, on the other, the $n$-fold suspension of the Thom space of $V$.

###### Definition

For $V \to X$ a vector bundle, its Thom spectrum is the Omega spectrum $E_\bullet$ with

$E_n \coloneqq (X^V)_n \coloneqq Th(\mathbb{R}^n \oplus V) \,.$

Without qualifiers, the Thom spectrum is that of the universal vector bundles:

###### Definition

For each $n \in \mathbb{N}$ let

$M O(n) := (B O)^{V(n)}$

be the Thom spectrum of the vector bundle $V(n)$ that is canonically associated to the $O(n)$-universal principal bundle $E O(n) \to B O(n)$ over the classifying space of the orthogonal group of dimension $n$.

The inclusions $O(n) \hookrightarrow O(n+1)$ induce a directed system of such spectra. The Thom spectrum is the colimit

$M O := {\lim_\to}_n M O(n) \,.$

Instead of the sequence of groups $O(n)$, one can consider $SO(n)$, or $Spin(n)$, $String(n)$, $Fivebrane(n)$,…, i.e., any level in the Whitehead tower of $O(n)$. To any of these groups there corresponds a Thom spectrum, which is in turn related to oriented cobordism, spin cobordism, string cobordism, et cetera.

(…)

### For $(\infty,1)$-module bundles

We discuss the Thom spectrum construction for general (∞,1)-module bundles.

###### Proposition

There is pair of adjoint (∞,1)-functors

$(\Sigma^\infty \Omega^\infty \dashv gl_1) : E_\infty Rings \stackrel{\overset{\Sigma^\infty \Omega^\infty}{\leftarrow}}{\underset{gl_1}{\to}} Spec_{con} \,,$

where $(\Sigma^\infty \dashv \Omega^\infty) : Spec \to Top$ is the stabilization adjunction between Top and Spec ($\Sigma^\infty$ forms the suspension spectrum), restricted to connective spectra. The right adjoint is the ∞-group of units-(∞,1)-functor, see there for more details.

This is (ABGHR, theorem 2.1/3.2).

###### Remark

Here $gl_1$ forms the “general linear group-of rank 1”-spectrum of an E-∞ ring: its ∞-group of units“. The adjunction is the generalization of the adjunction

$(\mathbb{Z}[-] \dashv GL_1) : CRing \stackrel{\overset{\mathbb{Z}[1]}{\leftarrow}}{\underset{GL_1}{\to}} Ab$

between CRing and Ab, where $\mathbb{Z}[-]$ forms the group ring.

###### Definition

Write

$b gl_1(R) := \Sigma gl_1(R)$

for the suspension of the group of units $gl_1(R)$.

This plays the role of the classifying space for $gl_1(R)$-principal ∞-bundles.

For $f : b \to b gl_1(R)$ a morphism (a cocycle for $gl_1(R)$-bundles) in Spec, write $p \to b$ for the corresponding bundle: the homotopy fiber

$\array{ p &\to& * \\ \downarrow && \downarrow \\ b &\stackrel{f}{\to}& b gl_1(R) } \,.$

Given a $R$-algebra $A$, hence an A-∞ algebra over $R$, exhibited by a morphism $\rho : R \to A$, the composite

$\rho(f) : b \stackrel{f}{\to} b gl_1(R) \stackrel{\rho}{\to} b gl_1(A)$

is that for the corresponding associated ∞-bundle.

We write capital letters for the underlying spaces of these spectra:

$P \coloneqq \Sigma^\infty \Omega^\infty p$
$B \coloneqq \Sigma^\infty \Omega^\infty b$
$GL_1(R) \coloneqq \Sigma^\infty \Omega^\infty gl_1(R)$
###### Definition

The Thom spectrum $M f$ of $f : b \to gl_1(R)$ is the (∞,1)-pushout

$\array{ \Sigma^\infty \Omega^\infty R &\to& R \\ \downarrow && \downarrow \\ \Sigma^\infty \Omega^\infty p &\to& M f } \,,$

hence the derived smash product

$M f \simeq P \wedge_{GL_1(R)} R \,.$
###### Remark

This means that a morphism $M f \to A$ is an $GL_1(R)$-equivariant map $P \to A$.

Notice that for $R = \mathbb{C}$ the complex numbers, $B \to GL_1(R)$ is the cocycle for a circle bundle $P \to B$. A $U(1)$-equivariant morphism $P \to A$ to some representation $A$ is equivalently a section of the A-associated bundle.

Therefore the Thom spectrum may be thought of as co-representing spaces of sections of associated bundles

$Hom(M f, A) \simeq \Gamma(P \wedge_{GL_1(R)} A)$”.

This is made precise by the following statement.

###### Proposition

We have an (∞,1)-pullback diagram

$\array{ E_\infty Alg_R(M f, A) &\stackrel{}{\to}& (...) \\ \downarrow && \downarrow \\ * &\stackrel{}{\to}& (...) }$

This is (ABGHR, theorem 2.10).

This definition does subsume the above definition of Thom spectra for sphere bundles (hence also that for vector bundles, by removing their zero section):

###### Proposition

Let $R = S$ be the sphere spectrum. Then for $f : b \to gl_1(S)$ a cocycle for an $S$-bundle,

$G := \Omega^\infty g : B \to B GL_1(S)$

is the classifying map for a spherical fibration over $B \in Top$.

The Thom spectrum $M f$ of def. 5 is equivalent to the Thom spectrum of the spherical fibration, according to def. 3.

This is in (ABGHR, section 8).

Equivalently the Thom spectrum is characterized as follows:

###### Definition/Proposition

For $\chi \colon X \to R Line$ a map to the ∞-group of $R$-(∞,1)-lines inside $R Mod$, the corresponding Thom spectrum is the (∞,1)-colimit

$\Gamma(\chi) \coloneqq \underset{\rightarrow}{\lim} \left( X \stackrel{\chi}{\to} Pic(R) \hookrightarrow R Mod \right) \,.$

This construction evidently extendes to an (∞,1)-functor

$\Gamma \colon \infty Grpd_{/R Line} \to R Mod \,.$

This is (Ando-Blumberg-Gepner 10, def. 4.1), reviewed also as (Wilson 13, def. 3.3).

###### Remark

This is the $R$-(∞,1)-module of sections of the (∞,1)-module bundle classified by $X \stackrel{\chi}{\to} Pic(R) \hookrightarrow R Mod$.

By the universal property of the (∞,1)-colimit we have for $\underline{R} \colon X \to R Mod$ the trivial $R$-bundle that

$Hom_{[X, R Mod]}(\chi, \underline{R}) \simeq Hom_{R Mod}(\Gamma(\chi), R) \,.$
###### Proposition

The section/Thom spectrum functor is the left (∞,1)-Kan extension of the canonical embedding $R Line \hookrightarrow R Mod$ along the (∞,1)-Yoneda embedding

$R Line \hookrightarrow [R Line^{op}, \infty Grpd] \simeq \infty Grpd_{/R Line}$

(where the equivalence of (∞,1)-categories on the right is given by the (∞,1)-Grothendieck construction). In other words, it is the essentially unique (∞,1)-colimit-preserving (∞,1)-functor $\infty Grpd_{/ R Line} \to R Mod$ which restricts along this inclusion to the canonical embedding.

This observation appears as (Wilson 13, prop. 4.4).

## Properties

### Relation to the cobordism ring

###### Proposition

The cobordism group of unoriented $n$-dimensional manifolds is naturally isomorphic to the $n$th homotopy group of the Thom spectrum $M O$. That is, there is a natural isomorphism

$\Omega^{un}_\bullet \simeq \pi_\bullet M O := {\lim_{\to}}_{k \to \infty} \pi_{n+k} M O(k) \,.$

This is a seminal result due to (Thom), whose proof proceeds by the Pontryagin-Thom construction. The presentation of the following proof follows (Francis, lecture 3).

###### Proof

We first construct a map $\Theta : \Omega_n^{un} \to \pi_n M O$.

Given a class $[X] \in \Omega_n^{un}$ we can choose a representative $X \in$ SmthMfd and a closed embedding $\nu$ of $X$ into the Cartesian space $\mathbb{R}^{n+k}$ of sufficiently large dimension. By the tubular neighbourhood theorem $\nu$ factors as the embedding of the zero section into the normal bundle $N_\nu$ followed by an open embedding of $N_\nu$ into $\mathbb{R}^{n+k}$

$\array{ X &&\stackrel{\nu}{\hookrightarrow}&& \mathbb{R}^{n+k} \\ & \searrow && \nearrow_{\mathrlap{i}} \\ && N_\nu } \,.$

Now use the Pontrjagin-Thom construction to produce an element of the homotopy group first in the Thom space $Th(N_\nu)$ of $N_\nu$ and then eventually in $M O$. To that end, let

$\mathbb{R}^{n+k} \to (\mathbb{R}^{n+k})^+ \simeq S^{n+k}$

be the map into the one-point compactification. Define a map

$t : S^{n+k} \simeq (\mathbb{R}^{n+k})^+ \to Disk(N_\nu)/Sphere(N_\nu) \simeq Th(N_\nu)$

by sending points in the image of $Disk(N_\nu)$ under $i$ to their preimage, and all other points to the collapsed point $Sphere(N_\nu)$. This defines an element in the homotopy group $\pi_{n+k}(Th(N_\nu))$.

To turn this into an element in the homotopy group of $M O$, notice that since $N_\nu$ is a vector bundle of rank $k$, it is the pullback by a map $\mu$ of the universal rank $k$ vector bundle $\gamma_k \to B O(k)$

$\array{ N_\nu \simeq \mu^* \gamma_k &\to& \gamma_k \\ \downarrow && \downarrow \\ X &\stackrel{\mu}{\to}& B O(k) } \,.$

By forming Thom spaces the top map induces a map

$Th(N_\nu) \to Th(\gamma^k) =: M O(k) \,.$

Its composite with the map $t$ constructed above gives an element in $\pi_{n+k} M O(k)$

$S^{n+k} \stackrel{t}{\to} Th(N_\nu) \to Th(\gamma^k) \simeq M O(k)$

and by $\pi_{n+k} M O(k) \to {\lim_\to}_k \pi_{n+k} M O(k) =: \pi_n M O$ this is finally an element

$\Theta : [X] \mapsto (S^{n+k} \stackrel{t}{\to} Th(N_\nu) \to Th(\gamma^k) \simeq M O(k)) \in \pi_n M O \,.$

We show now that this element does not depend on the choice of embedding $\nu : X \to \mathbb{R}^{n+k}$.

(…)

Finally, to show that $\Theta$ is an isomorphism by constructing an inverse.

For that, observe that the sphere $S^{n+k}$ is a compact topological space and in fact a compact object in Top. This implies that every map $f$ from $S^{n+k}$ into the filtered colimit

$Th(\gamma^k) \simeq {\lim_\to}_s Th(\gamma^k_s) \,,$

factors through one of the terms as

$f : S^{n+k} \to Th(\gamma^k_s) \hookrightarrow Th(\gamma^k) \,.$

By Thom's transversality theorem we may find an embedding $j : Gr_k(\mathbb{R}^s) \to Th(\Gamma^k_s)$ by a transverse map to $f$. Define then $X$ to be the pullback

$\array{ X &\to& Gr_k(\mathbb{R}^s) \\ \downarrow && \downarrow^{\mathrlap{j}} \\ S^{n+k} &\stackrel{f}{\to}& Th(\gamma^k_s) } \,.$

We check that this construction provides an inverse to $\Theta$.

(…)

###### Remark

The homotopy equivalence $\Omega^\infty M O \simeq \vert Cob_\infty \vert$ is the content of the Galatius-Madsen-Tillmann-Weiss theorem, and is now seen as a part of the cobordism hypothesis theorem.

### As a dual in the stable homotopy category

Write Spec for the category of spectra and $Ho(Spec)$ for its standard homotopy category: the stable homotopy category. By the symmetric monoidal smash product of spectra this becomes a monoidal category.

For $X$ any topological space, we may regard it as an object in $Ho(Spec)$ by forming its suspension spectrum $\Sigma^\infty_+ X$. We may ask under which conditions on $X$ this is a dualizable object with respect to the smash-product monoidal structure.

It turns out that a sufficient condition is that $X$ a closed smooth manifold or more generally a compact Euclidean neighbourhood retract. In that case $Th(N X)$ – the Thom spectrum of its stable normal bundle is the corresponding dual object. (Atiyah 61, Dold-Puppe 78). This is called the Spanier-Whitehead dual of $\Sigma^\infty_+ X$.

Using this one shows that the trace of the identity on $\Sigma^\infty_+ X$ in $Ho(Spec)$ – the categorical dimension of $\Sigma^\infty_+ X$ – is the Euler characteristic of $X$.

For a brief exposition see (PontoShulman, example 3.7). For more see at Spanier-Whitehead duality.

### As the universal spherical fibration, from the $J$-homomorphism

The J-homomorphism is a canonical map $B O \to B gl_1(\mathbb{S})$ from the classifying space of the stable orthogonal group to the delooping of the infinity-group of units of the sphere spectrum. This classifies an “(∞,1)-vector bundle” of sphere spectrum-modules over $B O$ and this is the Thom spectrum.

So in terms of the (∞,1)-colomit? description above we have

$M O \simeq \underset{\longrightarrow}{\lim}(B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to \mathbb{S}Mod = Spectra) \,.$

See at orientation in generalized cohomology for more on this.

### $E_\infty$-ring structure

Sufficient condition for a Thom spectrum to have E-∞ ring structure is that it arises, as above, as the (∞,1)-colimit of a homomorphism of E-∞ spaces $B \to A Line$ (ABG 10, prop.6.21).

### As the infinite cobordism category

The geometric realization for the (infinity,n)-category of cobordisms for $n \to \infty$ is the Thom spectrum

$\vert Bord_\infty \vert \simeq \Omega^\infty MO \,.$

This is implied by the Galatius-Madsen-Tillmann-Weiss theorem and by Jacob Lurie’s proof of the cobordism hypothesis. See also (Francis-Gwilliam, remark 0.9).

## Cohomology

Under the Brown representability theorem the Thom spectrum represents the generalized (Eilenberg-Steenrod) cohomology theory called cobordism cohomology theory.

The following terms all refer to essentially the same concept:

## References

### General

The relation between the homotopy groups of the Thom spectrum and the cobordism ring is due to

• René Thom, Quelques propriétés globales des variétés différentiables Comment. Math. Helv. 28, (1954). 17-86

Other original articles include

• Michael Atiyah, Thom complexes, Proc. London Math. Soc. (3), 11:291–310, 1961. 10

Lecture notes include

Textbook discussion with an eye towards the generalized (Eilenberg-Steenrod) cohomology of topological K-theory and cobordism cohomology theory is in

• Yuli Rudyak, On Thom spectra, orientability and cobordism, Springer Monographs in Mathematics, 1998 (pdf)

A generalized notion of Thom spectra in terms of (∞,1)-module bundles is discussed in

a streamlined update of which is

Discussion of Thom spectra from the point of view of (∞,1)-module bundles is in

which is reviewed in

• Dylan Wilson, Thom spectra from the $\infty$ point of view, 2013 (pdf)

and in the context of motivic quantization via pushforward in twisted generalized cohomology in section 3.1 of

### As dual objects in the stable homotopy category

The relation of Thom spectra to dualizable objects in the stable homotopy category is originally due to (Atiyah 61) and

• Albrecht Dold, Dieter Puppe, Duality, trace, and transfer. In Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), pages 81–102, Warsaw, 1980. PWN.
• L. G. Lewis, Jr., Peter May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, volume 1213 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure.

A brief exposition appears as example 3.7 in

Revised on May 20, 2014 14:43:59 by Toby Bartels (98.19.36.100)