nLab
Thom spectrum

Idea

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

$${\Omega }^{\mathrm{\infty }}MO\simeq \mid {\mathrm{Cob}}_{\mathrm{\infty }}\mid .$$\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

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

$$E_n≔(X^V{)}_n≔\mathrm{Th}({ℝ}^n\oplus V).$$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

$$MO(n):=(BO{)}^{V(n)}$$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 $EO(n)\to BO(n)$ 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

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

Instead of the sequence of groups $O(n)$, one can consider $\mathrm{SO}(n)$, or $\mathrm{Spin}(n)$, $\mathrm{String}(n)$, $\mathrm{Fivebrane}(n)$,…, 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 spherical fibrations

Definition

(…)

For $(\mathrm{\infty },1)$-bundles

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

Proposition

There is pair of adjoint functors

$$({\Sigma }^{\mathrm{\infty }}{\Omega }^{\mathrm{\infty }}⊣{\mathrm{gl}}_1):\mathrm{Ho}(E_{\mathrm{\infty }}\mathrm{Rings})\stackrel{\stackrel{{\Sigma }^{\mathrm{\infty }}{\Omega }^{\mathrm{\infty }}}{\leftarrow }}{\underset{{\mathrm{gl}}_1}{\to }}\mathrm{Ho}({\mathrm{Spec}}_{\mathrm{con}}),$$(\Sigma^\infty \Omega^\infty \dashv gl_1) : Ho(E_\infty Rings) \stackrel{\overset{\Sigma^\infty \Omega^\infty}{\leftarrow}}{\underset{gl_1}{\to}} Ho(Spec_{con}) \,,

where $({\Sigma }^{\mathrm{\infty }}⊣{\Omega }^{\mathrm{\infty }}):\mathrm{Spec}\to \mathrm{Top}$ is the stabilization adjunction between Top and Spec (${\Sigma }^{\mathrm{\infty }}$ forms the suspension spectrum), restricted to connective spectra, and $\mathrm{Ho}(-)$ denotes homotopy categories.

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 ”$\mathrm{\infty }$-group of units”. The adjunction is the generalization of the adjunction

$$(\mathbb{Z}[-]⊣{\mathrm{GL}}_1):\mathrm{CRing}\stackrel{\stackrel{\mathbb{Z}[1]}{\leftarrow }}{\underset{{\mathrm{GL}}_1}{\to }}\mathrm{Ab}$$(\mathbb{Z}[-] \dashv GL_1) : CRing \stackrel{\overset{\mathbb{Z}[1]}{\leftarrow}}{\underset{GL_1}{\to}} Ab

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

Definition

Write

$$b{\mathrm{gl}}_1(R):=\Sigma {\mathrm{gl}}_1(R)$$b gl_1(R) := \Sigma gl_1(R)

for the suspension of the group of units ${\mathrm{gl}}_1(R)$.

This plays the role of the classifying space for ${\mathrm{gl}}_1(R)$-principal ∞-bundles.

For $f:b\to b{\mathrm{gl}}_1(R)$ a morphism (a cocycle for ${\mathrm{gl}}_1(R)$-bundles) in Spec, write $p\to b$ for the corresponding bundle: the homotopy fiber

$$\begin{array}{ccc}p& \to & *\\ \downarrow & & \downarrow \\ b& \stackrel{f}{\to }& b{\mathrm{gl}}_1(R)\end{array}.$$\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

$$\rho (f):b\stackrel{f}{\to }b{\mathrm{gl}}_1(R)\stackrel{\rho }{\to }b{\mathrm{gl}}_1(A)$$\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:

$$P≔{\Sigma }^{\mathrm{\infty }}{\Omega }^{\mathrm{\infty }}p$$P \coloneqq \Sigma^\infty \Omega^\infty p

$$B≔{\Sigma }^{\mathrm{\infty }}{\Omega }^{\mathrm{\infty }}b$$B \coloneqq \Sigma^\infty \Omega^\infty b

$${\mathrm{GL}}_1(R)≔{\Sigma }^{\mathrm{\infty }}{\Omega }^{\mathrm{\infty }}{\mathrm{gl}}_1(R)$$GL_1(R) \coloneqq \Sigma^\infty \Omega^\infty gl_1(R)

Definition

The Thom spectrum $Mf$ of $f:b\to {\mathrm{gl}}_1(R)$ is the (∞,1)-pushout

$$\begin{array}{ccc}{\Sigma }^{\mathrm{\infty }}{\Omega }^{\mathrm{\infty }}R& \to & R\\ \downarrow & & \downarrow \\ {\Sigma }^{\mathrm{\infty }}{\Omega }^{\mathrm{\infty }}p& \to & Mf\end{array},$$\array{ \Sigma^\infty \Omega^\infty R &\to& R \\ \downarrow && \downarrow \\ \Sigma^\infty \Omega^\infty p &\to& M f } \,,

hence the derived smash product

$$Mf\simeq P{\wedge }_{{\mathrm{GL}}_1(R)}R.$$M f \simeq P \wedge_{GL_1(R)} R \,.

Remark

This means that a morphism $Mf\to A$ is an ${\mathrm{GL}}_1(R)$-equivariant map $P\to A$.

Notice that for $R=ℂ$ the complex numbers, $B\to {\mathrm{GL}}_1(R)$ 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

”$\mathrm{Hom}(Mf,A)\simeq \Gamma (P{\wedge }_{{\mathrm{GL}}_1(R)}A)$”.

This is made precise by the following statement.

Proposition

We have an (∞,1)-pullback diagram

$$\begin{array}{ccc}{E}_{\mathrm{\infty }}{\mathrm{Alg}}_R(Mf,A)& \stackrel{}{\to }& (...)\\ \downarrow & & \downarrow \\ *& \stackrel{}{\to }& (...)\end{array}$$\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):

Proposition

Let $R=S$ be the sphere spectrum. Then for $f:b\to {\mathrm{gl}}_1(S)$ a cocycle for an $S$-bundle,

$$G:={\Omega }^{\mathrm{\infty }}g:B\to B{\mathrm{GL}}_1(S)$$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).

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

$${\Omega }_{•}^{\mathrm{un}}\simeq {\pi }_{•}MO:={\underrightarrow{\lim }}_{k\to \mathrm{\infty }}{\pi }_{n+k}MO(k).$$\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 $[X]\in {\Omega }_n^{\mathrm{un}}$ 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 ${ℝ}^{n+k}$

$$\begin{array}{ccccc}X& & \stackrel{\nu }{\hookrightarrow }& & {ℝ}^{n+k}\\ & \searrow & & {\nearrow }_i\\ & & N_{\nu }\end{array}.$$\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 $\mathrm{Th}(N_{\nu })$ of $N_{\nu }$ and then eventually in $MO$. To that end, let

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

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

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

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 ${\gamma }_k\to BO(k)$

$$\begin{array}{ccc}{N}_{\nu }\simeq {\mu }^{*}{\gamma }_k& \to & {\gamma }_k\\ \downarrow & & \downarrow \\ X& \stackrel{\mu }{\to }& BO(k)\end{array}.$$\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

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

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

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

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

$$\Theta :[X]\mapsto (S^{n+k}\stackrel{t}{\to }\mathrm{Th}(N_{\nu })\to \mathrm{Th}({\gamma }^k)\simeq MO(k))\in {\pi }_{n}MO.$$\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

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

factors through one of the terms as

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

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

$$\begin{array}{ccc}X& \to & {\mathrm{Gr}}_k({ℝ}^s)\\ \downarrow & & {\downarrow }^j\\ S^{n+k}& \stackrel{f}{\to }& \mathrm{Th}({\gamma }_s^k)\end{array}.$$\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 ${\Omega }^{\mathrm{\infty }}MO\simeq \mid {\mathrm{Cob}}_{\mathrm{\infty }}\mid$ 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 $\mathrm{Ho}(\mathrm{Spec})$ 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 $\mathrm{Ho}(\mathrm{Spec})$ by forming its suspension spectrum ${\Sigma }_{+}^{\mathrm{\infty }}X$. 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. In that case $\mathrm{Th}(NX)$ – the Thom spectrum of its stable normal bundle is the corresponding dual object. This is called the Spanier-Whitehead dual of ${\Sigma }_{+}^{\mathrm{\infty }}X$.

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

This characterization is due to Dold. It is mentioned as an example of traces in the expository (PontoShulman, example 3.7). For more see Spanier-Whitehead duality.

Cohomology

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

Related concepts

• Thom space, Thom isomorphism

• cobordism ring

References

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

A review is in

• John Francis, Topology of manifolds course notes (2010) (web)

  Lecture 3 Thom’s theorem (notes by A. Smith) (pdf)

  A remark of the relation of the Thom spectrum to (∞,n)-category of cobordisms for $n=\mathrm{\infty }$ is in:

  Lecture 2 Cobordisms (notes by Owen Gwilliam) (pdf)

The generalized notion of Thom spectra is discussed in

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)

The relation of Thom spectra to dualizable objects in the stable homotopy category is mentioned as example 3.7 in the expository

• Kate Ponto and Mike Shulman, Traces in symmetric monoidal categories (pdf, slides).