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Thom spectrum

Contents

Idea

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

Ω MO|Cob |. \Omega^\infty M O \simeq \vert Cob_\infty \vert .

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

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

MOlim(BOJBGL 1(𝕊)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 BOB O for (stable) real vector bundles.

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

MGlim(BGBOJBGL 1(𝕊)Spectra). 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 GG in the Whitehead tower of the orthogonal group, where it yields MMSpin, MMString group, etc.

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

Definition

For vector bundles

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

Th( nV)S nTh(V)Σ nTh(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 VV with the trivial vector bundle of rank nn and, on the other, the nn-fold suspension of the Thom space of VV.

Definition

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

E n(X V) nTh( nV). 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 nn \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)V(n) that is canonically associated to the O(n)O(n)-universal principal bundle EO(n)BO(n)E O(n) \to B O(n) over the classifying space of the orthogonal group of dimension nn.

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

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

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

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

Proposition

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

(Σ Ω gl 1):E Ringsgl 1Σ Ω Spec con, (\Sigma^\infty \Omega^\infty \dashv gl_1) : E_\infty Rings \stackrel{\overset{\Sigma^\infty \Omega^\infty}{\leftarrow}}{\underset{gl_1}{\to}} Spec_{con} \,,

where (Σ Ω ):SpecTop(\Sigma^\infty \dashv \Omega^\infty) : Spec \to Top is the stabilization adjunction between Top and Spec (Σ \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 1gl_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

([]GL 1):CRingGL 1[1]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

bgl 1(R):=Σgl 1(R) b gl_1(R) := \Sigma gl_1(R)

for the suspension of the group of units gl 1(R)gl_1(R).

This plays the role of the classifying space for gl 1(R)gl_1(R)-principal ∞-bundles.

For f:bbgl 1(R)f : b \to b gl_1(R) a morphism (a cocycle for gl 1(R)gl_1(R)-bundles) in Spec, write pbp \to b for the corresponding bundle: the homotopy fiber

p * b f bgl 1(R). \array{ p &\to& * \\ \downarrow && \downarrow \\ b &\stackrel{f}{\to}& b gl_1(R) } \,.

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

ρ(f):bfbgl 1(R)ρbgl 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:

PΣ Ω p P \coloneqq \Sigma^\infty \Omega^\infty p
BΣ Ω b B \coloneqq \Sigma^\infty \Omega^\infty b
GL 1(R)Σ Ω gl 1(R) GL_1(R) \coloneqq \Sigma^\infty \Omega^\infty gl_1(R)
Definition

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

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

hence the derived smash product

MfP GL 1(R)R. M f \simeq P \wedge_{GL_1(R)} R \,.
Remark

This means that a morphism MfAM f \to A is an GL 1(R)GL_1(R)-equivariant map PAP \to A.

Notice that for R=R = \mathbb{C} the complex numbers, BGL 1(R)B \to GL_1(R) is the cocycle for a circle bundle PBP \to B. A U(1)U(1)-equivariant morphism PAP \to A to some representation AA 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(Mf,A)Γ(P GL 1(R)A)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

E Alg R(Mf,A) (...) * (...) \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=SR = S be the sphere spectrum. Then for f:bgl 1(S)f : b \to gl_1(S) a cocycle for an SS-bundle,

G:=Ω g:BBGL 1(S) G := \Omega^\infty g : B \to B GL_1(S)

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

The Thom spectrum MfM 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 χ:XRLine\chi \colon X \to R Line a map to the ∞-group of RR-(∞,1)-lines inside RModR Mod, the corresponding Thom spectrum is the (∞,1)-colimit

Γ(χ)lim(XχPic(R)RMod). \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

Γ:Grpd /RLineRMod. \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 RR-(∞,1)-module of sections of the (∞,1)-module bundle classified by XχPic(R)RModX \stackrel{\chi}{\to} Pic(R) \hookrightarrow R Mod.

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

Hom [X,RMod](χ,R̲)Hom RMod(Γ(χ),R). 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 RLineRModR Line \hookrightarrow R Mod along the (∞,1)-Yoneda embedding

RLine[RLine op,Grpd]Grpd /RLine 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 Grpd /RLineRMod\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 nn-dimensional manifolds is naturally isomorphic to the nnth homotopy group of the Thom spectrum MOM O. That is, there is a natural isomorphism

Ω unπ MO:=lim kπ n+kMO(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 Θ:Ω n unπ nMO\Theta : \Omega_n^{un} \to \pi_n M O.

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

X ν n+k i N ν. \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 ν)Th(N_\nu) of N νN_\nu and then eventually in MOM O. To that end, let

n+k( n+k) +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( n+k) +Disk(N ν)/Sphere(N ν)Th(N ν) 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 ν)Disk(N_\nu) under ii to their preimage, and all other points to the collapsed point Sphere(N ν)Sphere(N_\nu). This defines an element in the homotopy group π n+k(Th(N ν))\pi_{n+k}(Th(N_\nu)).

To turn this into an element in the homotopy group of MOM O, notice that since N νN_\nu is a vector bundle of rank kk, it is the pullback by a map μ\mu of the universal rank kk vector bundle γ kBO(k)\gamma_k \to B O(k)

N νμ *γ k γ k X μ BO(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 ν)Th(γ k)=:MO(k). Th(N_\nu) \to Th(\gamma^k) =: M O(k) \,.

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

S n+ktTh(N ν)Th(γ k)MO(k) S^{n+k} \stackrel{t}{\to} Th(N_\nu) \to Th(\gamma^k) \simeq M O(k)

and by π n+kMO(k)lim kπ n+kMO(k)=:π nMO\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

Θ:[X](S n+ktTh(N ν)Th(γ k)MO(k))π nMO. \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 ν:X n+k\nu : X \to \mathbb{R}^{n+k}.

(…)

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

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

Th(γ k)lim sTh(γ s k), Th(\gamma^k) \simeq {\lim_\to}_s Th(\gamma^k_s) \,,

factors through one of the terms as

f:S n+kTh(γ s k)Th(γ 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:Gr k( s)Th(Γ s k)j : Gr_k(\mathbb{R}^s) \to Th(\Gamma^k_s) by a transverse map to ff. Define then XX to be the pullback

X Gr k( s) j S n+k f Th(γ s k). \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 Θ\Theta.

(…)

Remark

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

It turns out that a sufficient condition is that XX a closed smooth manifold or more generally a compact Euclidean neighbourhood retract. In that case Th(NX)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 Σ + X\Sigma^\infty_+ X.

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

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

As the universal spherical fibration, from the JJ-homomorphism

The J-homomorphism is a canonical map BOBgl 1(𝕊)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 BOB O and this is the Thom spectrum.

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

MOlim(BOJBGL 1(𝕊)𝕊Mod=Spectra). 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 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 BALineB \to A Line (ABG 10, prop.6.21).

As the infinite cobordism category

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

|Bord |Ω MO. \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

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)