# nLab Thom spectrum

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

## Theorem

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

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

${\Omega }^{\infty }MO\simeq \mid {\mathrm{Cob}}_{\infty }\mid .$\Omega^\infty M O \simeq \vert Cob_\infty \vert .

In particular, ${\pi }_{n}MO$ is naturally identified with the set of cobordism classes of closed $n$-manifolds.

## Definition

### For vector bundles

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

$\mathrm{Th}\left({ℝ}^{n}\oplus V\right)\simeq {S}^{n}\wedge \mathrm{Th}\left(V\right)\simeq {\Sigma }^{n}\mathrm{Th}\left(V\right)$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 ${E}_{•}$ with

${E}_{n}≔\left({X}^{V}{\right)}_{n}≔\mathrm{Th}\left({ℝ}^{n}\oplus V\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $n\in ℕ$ let

$MO\left(n\right):=\left(BO{\right)}^{V\left(n\right)}$M O(n) := (B O)^{V(n)}

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

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

$MO:={\underset{\to }{\mathrm{lim}}}_{n}MO\left(n\right)\phantom{\rule{thinmathspace}{0ex}}.$M O := {\lim_\to}_n M O(n) \,.

Instead of the sequence of groups $O\left(n\right)$, one can consider $\mathrm{SO}\left(n\right)$, or $\mathrm{Spin}\left(n\right)$, $\mathrm{String}\left(n\right)$, $\mathrm{Fivebrane}\left(n\right)$,…, i.e., any level in the Whitehead tower of $O\left(n\right)$. 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 $\left(\infty ,1\right)$-module bundles

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

###### Proposition

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

$\left({\Sigma }^{\infty }{\Omega }^{\infty }⊣{\mathrm{gl}}_{1}\right):{E}_{\infty }\mathrm{Rings}\stackrel{\stackrel{{\Sigma }^{\infty }{\Omega }^{\infty }}{←}}{\underset{{\mathrm{gl}}_{1}}{\to }}{\mathrm{Spec}}_{\mathrm{con}}\phantom{\rule{thinmathspace}{0ex}},$(\Sigma^\infty \Omega^\infty \dashv gl_1) : E_\infty Rings \stackrel{\overset{\Sigma^\infty \Omega^\infty}{\leftarrow}}{\underset{gl_1}{\to}} Spec_{con} \,,

where $\left({\Sigma }^{\infty }⊣{\Omega }^{\infty }\right):\mathrm{Spec}\to \mathrm{Top}$ 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 ${\mathrm{gl}}_{1}$ 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

$\left(ℤ\left[-\right]⊣{\mathrm{GL}}_{1}\right):\mathrm{CRing}\stackrel{\stackrel{ℤ\left[1\right]}{←}}{\underset{{\mathrm{GL}}_{1}}{\to }}\mathrm{Ab}$(\mathbb{Z}[-] \dashv GL_1) : CRing \stackrel{\overset{\mathbb{Z}[1]}{\leftarrow}}{\underset{GL_1}{\to}} Ab

between CRing and Ab, where $ℤ\left[-\right]$ forms the group ring.

###### Definition

Write

$b{\mathrm{gl}}_{1}\left(R\right):=\Sigma {\mathrm{gl}}_{1}\left(R\right)$b gl_1(R) := \Sigma gl_1(R)

for the suspension of the group of units ${\mathrm{gl}}_{1}\left(R\right)$.

This plays the role of the classifying space for ${\mathrm{gl}}_{1}\left(R\right)$-principal ∞-bundles.

For $f:b\to b{\mathrm{gl}}_{1}\left(R\right)$ a morphism (a cocycle for ${\mathrm{gl}}_{1}\left(R\right)$-bundles) in Spec, write $p\to b$ for the corresponding bundle: the homotopy fiber

$\begin{array}{ccc}p& \to & *\\ ↓& & ↓\\ b& \stackrel{f}{\to }& b{\mathrm{gl}}_{1}\left(R\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\rho \left(f\right):b\stackrel{f}{\to }b{\mathrm{gl}}_{1}\left(R\right)\stackrel{\rho }{\to }b{\mathrm{gl}}_{1}\left(A\right)$\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:

$P≔{\Sigma }^{\infty }{\Omega }^{\infty }p$P \coloneqq \Sigma^\infty \Omega^\infty p
$B≔{\Sigma }^{\infty }{\Omega }^{\infty }b$B \coloneqq \Sigma^\infty \Omega^\infty b
${\mathrm{GL}}_{1}\left(R\right)≔{\Sigma }^{\infty }{\Omega }^{\infty }{\mathrm{gl}}_{1}\left(R\right)$GL_1(R) \coloneqq \Sigma^\infty \Omega^\infty gl_1(R)
###### Definition

The Thom spectrum $Mf$ of $f:b\to {\mathrm{gl}}_{1}\left(R\right)$ is the (∞,1)-pushout

$\begin{array}{ccc}{\Sigma }^{\infty }{\Omega }^{\infty }R& \to & R\\ ↓& & ↓\\ {\Sigma }^{\infty }{\Omega }^{\infty }p& \to & Mf\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ \Sigma^\infty \Omega^\infty R &\to& R \\ \downarrow && \downarrow \\ \Sigma^\infty \Omega^\infty p &\to& M f } \,,

hence the derived smash product

$Mf\simeq P{\wedge }_{{\mathrm{GL}}_{1}\left(R\right)}R\phantom{\rule{thinmathspace}{0ex}}.$M f \simeq P \wedge_{GL_1(R)} R \,.
###### Remark

This means that a morphism $Mf\to A$ is an ${\mathrm{GL}}_{1}\left(R\right)$-equivariant map $P\to A$.

Notice that for $R=ℂ$ the complex numbers, $B\to {\mathrm{GL}}_{1}\left(R\right)$ is the cocycle for a circle bundle $P\to B$. A $U\left(1\right)$-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

$\mathrm{Hom}\left(Mf,A\right)\simeq \Gamma \left(P{\wedge }_{{\mathrm{GL}}_{1}\left(R\right)}A\right)$”.

This is made precise by the following statement.

###### Proposition

We have an (∞,1)-pullback diagram

$\begin{array}{ccc}{E}_{\infty }{\mathrm{Alg}}_{R}\left(Mf,A\right)& \stackrel{}{\to }& \left(...\right)\\ ↓& & ↓\\ *& \stackrel{}{\to }& \left(...\right)\end{array}$\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 $f:b\to {\mathrm{gl}}_{1}\left(S\right)$ a cocycle for an $S$-bundle,

$G:={\Omega }^{\infty }g:B\to B{\mathrm{GL}}_{1}\left(S\right)$G := \Omega^\infty g : B \to B GL_1(S)

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

The Thom spectrum $Mf$ 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 $\chi :X\to R\mathrm{Line}$ a map to the ∞-group of $R$-(∞,1)-lines inside $R\mathrm{Mod}$, the corresponding Thom spectrum is the (∞,1)-colimit

$\Gamma \left(\chi \right)≔\underset{\to }{\mathrm{lim}}\left(X\stackrel{\chi }{\to }\mathrm{Pic}\left(R\right)↪R\mathrm{Mod}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\Gamma :\infty {\mathrm{Grpd}}_{/R\mathrm{Line}}\to R\mathrm{Mod}\phantom{\rule{thinmathspace}{0ex}}.$\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 $X\stackrel{\chi }{\to }\mathrm{Pic}\left(R\right)hookrightarowR\mathrm{Mod}$.

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

${\mathrm{Hom}}_{\left[X,R\mathrm{Mod}\right]}\left(\chi ,\underline{R}\right)\simeq {\mathrm{Hom}}_{R\mathrm{Mod}}\left(\Gamma \left(\chi \right),R\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $R\mathrm{Line}↪R\mathrm{Mod}$ along the (∞,1)-Yoneda embedding

$R\mathrm{Line}↪\left[R{\mathrm{Line}}^{\mathrm{op}},\infty \mathrm{Grpd}\right]\simeq \infty {\mathrm{Grpd}}_{/R\mathrm{Line}}$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 $\infty {\mathrm{Grpd}}_{/R\mathrm{Line}}\to R\mathrm{Mod}$ 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 $MO$. That is, there is a natural isomorphism

${\Omega }_{•}^{\mathrm{un}}\simeq {\pi }_{•}MO:={\underset{\to }{\mathrm{lim}}}_{k\to \infty }{\pi }_{n+k}MO\left(k\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $\Theta :{\Omega }_{n}^{\mathrm{un}}\to {\pi }_{n}MO$.

Given a class $\left[X\right]\in {\Omega }_{n}^{\mathrm{un}}$ we can choose a representative $X\in$ SmthMfd and a closed embedding $\nu$ of $X$ into the Cartesian space ${ℝ}^{n+k}$ 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 ${ℝ}^{n+k}$

$\begin{array}{ccccc}X& & \stackrel{\nu }{↪}& & {ℝ}^{n+k}\\ & ↘& & {↗}_{i}\\ & & {N}_{\nu }\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\mathrm{Th}\left({N}_{\nu }\right)$ of ${N}_{\nu }$ and then eventually in $MO$. To that end, let

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

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

$t:{S}^{n+k}\simeq \left({ℝ}^{n+k}{\right)}^{+}\to \mathrm{Disk}\left({N}_{\nu }\right)/\mathrm{Sphere}\left({N}_{\nu }\right)\simeq \mathrm{Th}\left({N}_{\nu }\right)$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 $\mathrm{Disk}\left({N}_{\nu }\right)$ under $i$ to their preimage, and all other points to the collapsed point $\mathrm{Sphere}\left({N}_{\nu }\right)$. This defines an element in the homotopy group ${\pi }_{n+k}\left(\mathrm{Th}\left({N}_{\nu }\right)\right)$.

To turn this into an element in the homotopy group of $MO$, 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 ${\gamma }_{k}\to BO\left(k\right)$

$\begin{array}{ccc}{N}_{\nu }\simeq {\mu }^{*}{\gamma }_{k}& \to & {\gamma }_{k}\\ ↓& & ↓\\ X& \stackrel{\mu }{\to }& BO\left(k\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\mathrm{Th}\left({N}_{\nu }\right)\to \mathrm{Th}\left({\gamma }^{k}\right)=:MO\left(k\right)\phantom{\rule{thinmathspace}{0ex}}.$Th(N_\nu) \to Th(\gamma^k) =: M O(k) \,.

Its composite with the map $t$ constructed above gives an element in ${\pi }_{n+k}MO\left(k\right)$

${S}^{n+k}\stackrel{t}{\to }\mathrm{Th}\left({N}_{\nu }\right)\to \mathrm{Th}\left({\gamma }^{k}\right)\simeq MO\left(k\right)$S^{n+k} \stackrel{t}{\to} Th(N_\nu) \to Th(\gamma^k) \simeq M O(k)

and by ${\pi }_{n+k}MO\left(k\right)\to {{\mathrm{lim}}_{\to }}_{k}{\pi }_{n+k}MO\left(k\right)=:{\pi }_{n}MO$ this is finally an element

$\Theta :\left[X\right]↦\left({S}^{n+k}\stackrel{t}{\to }\mathrm{Th}\left({N}_{\nu }\right)\to \mathrm{Th}\left({\gamma }^{k}\right)\simeq MO\left(k\right)\right)\in {\pi }_{n}MO\phantom{\rule{thinmathspace}{0ex}}.$\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 $\nu :X\to {ℝ}^{n+k}$.

(…)

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

$\mathrm{Th}\left({\gamma }^{k}\right)\simeq {\underset{\to }{\mathrm{lim}}}_{s}\mathrm{Th}\left({\gamma }_{s}^{k}\right)\phantom{\rule{thinmathspace}{0ex}},$Th(\gamma^k) \simeq {\lim_\to}_s Th(\gamma^k_s) \,,

factors through one of the terms as

$f:{S}^{n+k}\to \mathrm{Th}\left({\gamma }_{s}^{k}\right)↪\mathrm{Th}\left({\gamma }^{k}\right)\phantom{\rule{thinmathspace}{0ex}}.$f : S^{n+k} \to Th(\gamma^k_s) \hookrightarrow Th(\gamma^k) \,.

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

$\begin{array}{ccc}X& \to & {\mathrm{Gr}}_{k}\left({ℝ}^{s}\right)\\ ↓& & {↓}^{j}\\ {S}^{n+k}& \stackrel{f}{\to }& \mathrm{Th}\left({\gamma }_{s}^{k}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 ${\Omega }^{\infty }MO\simeq \mid {\mathrm{Cob}}_{\infty }\mid$ 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 $\mathrm{Ho}\left(\mathrm{Spec}\right)$ 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 $\mathrm{Ho}\left(\mathrm{Spec}\right)$ 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 $\mathrm{Th}\left(NX\right)$ – 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 $\mathrm{Ho}\left(\mathrm{Spec}\right)$ – the categorical dimension of ${\Sigma }_{+}^{\infty }X$ – is the Euler characteristic of $X$.

For an 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 $BO\to B{\mathrm{gl}}_{1}\left(𝕊\right)$ 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 $BO$ and this is the Thom spectrum.

See at orientation in generalized cohomology for more on this.

### As the infinite cobordism category

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

$\mid {\mathrm{Bord}}_{\infty }\mid \simeq {\Omega }^{\infty }\mathrm{MO}\phantom{\rule{thinmathspace}{0ex}}.$\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

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 December 9, 2013 13:25:48 by Urs Schreiber (89.204.139.72)