nLab Thom spectrum



Cobordism theory

Stable Homotopy theory





Universal Thom spectra

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 (Thom's theorem).

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 B O B 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).


For vector bundles

First recall the following two basic facts about the construction of Thom spaces. See at Thom space this prop..


For VXV \to X a real vector bundle, there is a homeomorphism

Th( nV)S nTh(V)=Σ nTh(V) Th(\mathbb{R}^n \oplus V) \simeq S^n \wedge Th(V) = \Sigma^n Th(V)

between, on the one hand, the Thom space of the direct sum of vector bundles of VV with the trivial vector bundle of rank nn and, on the other, the nn-fold reduced suspension of the Thom space of VV.


For V 1X 1V_1 \to X_1 and V 2X 2V_2 \to X_2 two real vector bundles, let V 1V 2X 1×X 2V_1 \boxtimes V_2 \to X_1 \times X_2 be the direct sum of vector bundles of their pullbacks to X 1×X 2X_1 \times X_2 (their external tensor product). The corresponding Thom space is the smash product of the individual Thom spaces:

Th(V 1V 2)Th(V 1)Th(V 2). Th(V_1 \boxtimes V_2) \simeq Th(V_1) \wedge Th(V_2) \,.

Prop. will give rise to universal Thom spectra in the following, while Prop. will give them the struture of ring spectra.


For VXV \to X a real vector bundle, its Thom spectrum is the suspension spectrum Σ Th(V)\Sigma^\infty Th(V) of the Thom space of VV. By prop. this may be written as

(Σ Th(V)) nTh( nV) (\Sigma^\infty Th(V))_n \;\simeq\; Th(\mathbb{R}^n \oplus V)

with structure maps (as a sequential spectrum) the equivalences

σ n:ΣTh( nV)Th( n+1V). \sigma_n \;\colon\; \Sigma Th(\mathbb{R}^n \oplus V) \stackrel{\simeq}{\longrightarrow} Th(\mathbb{R}^{n+1} \oplus V) \,.

For each nn \in \mathbb{N} the pullback of the rank-(n+1)(n+1) universal vector bundle to the classifying space of rank nn vector bundles is the direct sum of vector bundles of the rank nn universal vector bundle with the trivial rank-1 bundle: there is a pullback diagram in Top of the form

(EO(n)×O(n) n) EO(n+1)×O(n+1) n+1 (pb) BO(n) BO(n+1), \array{ \mathbb{R}\oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^n) &\longrightarrow& E O(n+1) \underset{O(n+1)}{\times} \mathbb{R}^{n+1} \\ \big\downarrow &(pb)& \big\downarrow \\ B O(n) &\longrightarrow& B O(n+1) } \,,

where the bottom morphism is the canonical one (this def.).

(e.g. Kochmann 96, p. 25)


For each kk \in \mathbb{N}, knk \geq n there are such pullbacks of the canonical vector bundles over Grassmannians

{V n k,vV n,v n+1} {V n+1 k+1,vV n+1} Gr n( k) Gr n+1( k+1) \array{ \left\{ {V_{n}\subset \mathbb{R}^k, } \atop {v \in V_n, v_{n+1} \in \mathbb{R}} \right\} &\longrightarrow& \left\{ {V_{n+1} \subset \mathbb{R}^{k+1}}, \atop v \in V_{n+1} \right\} \\ \downarrow && \downarrow \\ Gr_n(\mathbb{R}^k) &\longrightarrow& Gr_{n+1}(\mathbb{R}^{k+1}) }

where the bottom morphism is the canonical inclusion (def.). Under taking the colimit over kk, this produces the claimed pullback.


The nn-fold looping of the Thom spectrum, according to def. , of the rank-nn universal vector bundle is written

MO(n)Σ nΣ Th(EO(n)×O(n) n). M O(n) \coloneqq \Sigma^{-n} \Sigma^\infty Th( E O(n) \underset{O(n)}{\times} \mathbb{R}^n ) \,.

The image of the top horizontal maps in prop. under Σ nTh()\Sigma^{\infty - n}Th(-) are, via prop. , maps of the form

MO(n)MO(n+1) M O(n) \longrightarrow M O(n+1)

The homotopy colimit over these maps is the universal Thom spectrum:

MOlim nMO(n). M O \coloneqq {\lim_\to}_n M O(n) \,.

More explicitly:


As a sequential spectrum, the universal Thom spectrum MOM O is represented by the sequential prespectrum whose nnth component space is the Thom space

(MO) nTh(EO(n)×O(n) n) (M O)_n \coloneqq Th(E O(n)\underset{O(n)}{\times}\mathbb{R}^n)

of the rank-nn universal vector bundle, and whose structure maps are the image under the Thom space functor Th()Th(-) of the top morphisms in prop. , via the homeomorphisms of prop. :

σ n:Σ(MO) nTh((EO(n)×O(n) n))Th(EO(n+1)×O(n+1) n+1)=(MO) n+1. \sigma_n \;\colon\; \Sigma (M O)_n \simeq Th(\mathbb{R}\oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^n)) \stackrel{}{\longrightarrow} Th(E O(n+1) \underset{O(n+1)}{\times} \mathbb{R}^{n+1}) = (M O)_{n+1} \,.

As an orthogonal ring spectrum, the universal Thom spectrum MOM O has

  • component spaces

    (MO) VEO(V) +O(V)S V (M O)_V \coloneqq E O(V)_+ \underset{O(V)}{\wedge} S^V

    the Thom spaces of the universal vector bundle with fiber VV;

  • left O(V)O(V)-action induced by the remaining canonical left action of EO(V)E O(V);

  • multiplication maps

    (EO(V 1) +O(V 1)S V 1)(EO(V 2) +O(V 2)S V 2EO(V 1V 2) +O(V 1V 2)S V 1V 2 (E O(V_1)_+ \underset{O(V_1)}{\wedge} S^{V_1}) \wedge (E O(V_2)_+ \underset{O(V_2)}{\wedge} S^{V_2} \longrightarrow E O(V_1 \oplus V_2)_+ \underset{O(V_1 \oplus V_2)}{\wedge} S^{V_1 \oplus V_2}

    induced via prop.

  • unit maps given by

    S VO(V) + O(V)S VEO(V) + O(V)S V S^V \simeq O(V)_+ \wedge_{O(V)} S^V \longrightarrow E O(V)_+ \wedge_{O(V)} S^V

    induced by the fiber inclusion O(V)EO(V)O(V) \to E O(V).

For discussion of the refinement of the Thom spectrum MOM O to a symmetric spectrum see (Hovey-Shipley-Smith 00, example 6.2.3, Schwede 12, Example I.2.8). For the refinement to an orthogonal spectrum and globally equivariant spectrum see (Schwede 15, section V.4).

More generally, there are universal Thom spectra associated with any other tangent structure (“(B,f)-structure”), notably for the orthogonal group replaced by the special orthogonal groups SO(n)SO(n), or the spin groups Spin(n)Spin(n), or the string 2-group String(n)String(n), or the fivebrane 6-group Fivebrane(n)Fivebrane(n),…, or any level in the Whitehead tower of O(n)O(n). To any of these groups there corresponds a Thom spectrum (denoted, respectively, MSOM SO, MSpinM Spin, MStringM String, MFivebraneM Fivebrane, etc.), which is in turn related to oriented cobordism, spin cobordism, string cobordism, et cetera.



A (B,f)-structure \mathcal{B}, is a system of Serre fibrations

B n f n BO(n) \array{ B_n \\ \downarrow^{\mathrlap{f_n}} \\ B O(n) }

for nn \in \mathbb{N}, equipped with maps j n:B nB n+1j_n \colon B_n \to B_{n+1} covering the canonical maps i n:BO(n)BO(n+1)i_n \colon B O(n) \to B O(n+1) ((def.)) in that there are commuting squares

B n j n B n+1 f n f n+1 BO(n) i n BO(n+1). \array{ B_n &\overset{j_n}{\longrightarrow}& B_{n+1} \\ \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{f_{n+1}}} \\ B O(n) &\overset{i_n}{\longrightarrow}& B O(n+1) } \,.

Similarly, an S 2S^2-(B,f)(B,f)-structure is a compatible system

f 2n:B 2nBO(2n) f_{2n} \colon B_{2n} \longrightarrow B O(2n)

indexed only on the even natural numbers.


Given a (B,f)-structure \mathcal{B} (def. ), write

E n f n *(EO(n)×O(n) n) E^{\mathcal{B}}_n \coloneqq f_n^\ast ( E O(n) \underset{O(n)}{\times} \mathbb{R}^n )

for the pullback of the rank-nn universal vector bundle from B O ( n ) B O(n) to B nB_n along f nf_n.

Observe that the analog of prop. still holds:


Given a (B,f)-structure \mathcal{B} (def. ), then the pullback of its rank-(n+1)(n+1) vector bundle E n+1 E^{\mathcal{B}}_{n+1} (def. ) along the map j n:B nB n+1j_n \colon B_n \to B_{n+1} is the direct sum of vector bundles of the rank-nn bundle E n E^{\mathcal{B}}_n with the trivial rank-1-bundle: there is a pullback square

E n E n+1 (pb) B n j n B n+1. \array{ \mathbb{R} \oplus E^{\mathcal{B}}_n &\longrightarrow& E^{\mathcal{B}}_{n+1} \\ \downarrow &(pb)& \downarrow \\ B_n &\underset{j_n}{\longrightarrow}& B_{n+1} } \,.

Unwinding the definitions, the pullback in question is

j n *E n+1 =j n *f n+1 *(EO(n+1)×O(n+1) n+1) (j nf n+1) *(EO(n+1)×O(n+1) n+1) (f ni n) *(EO(n+1)×O(n+1) n+1) f n *i n *(EO(n+1)×O(n+1) n+1) f n *((EO(n)×O(n) n)) E n, \begin{aligned} j_n^\ast E^{\mathcal{B}}_{n+1} & = j_n^\ast f_{n+1}^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq (j_n \circ f_{n+1})^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq ( f_n \circ i_n )^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq f_n^\ast i_n^\ast (E O(n+1)\underset{O(n+1)}{\times} \mathbb{R}^{n+1}) \\ & \simeq f_n^\ast (\mathbb{R} \oplus (E O(n)\underset{O(n)}{\times} \mathbb{R}^{n})) \\ &\simeq \mathbb{R} \oplus E^{\mathcal{B}_n} \,, \end{aligned}

where the second but last step is due to prop. .


Given a (B,f)-structure \mathcal{B} (def. ), its Thom spectrum MM \mathcal{B} is, as a sequential prespectrum, given by component spaces being the Thom spaces of the \mathcal{B}-associated vector bundles of def.

(M) nTh(E n ) (M \mathcal{B})_n \coloneqq Th(E^{\mathcal{B}}_n)

and with structure maps given via prop. by the top maps in prop. :

σ n:Σ(M) n=ΣTh(E n )Th(E n )Th(E n+1 )=(M) n+1. \sigma_n \;\colon\; \Sigma (M \mathcal{B})_n = \Sigma Th(E^{\mathcal{E}}_n) \simeq Th(\mathbb{R}\oplus E^{\mathcal{E}}_n) \longrightarrow Th(E^{\mathcal{E}}_{n+1}) = (M \mathcal{B})_{n+1} \,.

Similarly for a (B,f)(B,f)-structure indexed on the even natural numbers, there is the corresponding Thom spectrum as an S 2S^2-sequential spectrum (def.).

If B n=BG nB_n = B G_n for some natural system of groups G nO(n)G_n \to O(n), then one usually writes MGM G for MM \mathcal{B}. For instance MSOM SO, MSpinM Spin, MUM U etc.

If the (B,f)(B,f)-structure is multiplicative (def. ), then the Thom spectrum MM \mathcal{B} canonical becomes a ring spectrum: the multiplication maps B n 1×B n 2B n 1+n 2B_{n_1} \times B_{n_2}\to B_{n_1 + n_2} are covered by maps of vector bundles

E n 1 E n 2 E n 1+n 2 E^{\mathcal{B}}_{n_1} \boxtimes E^{\mathcal{B}}_{n_2} \longrightarrow E^{\mathcal{B}}_{n_1 + n_2}

and under forming Thom spaces this yields, by prop. , maps

(M) n 1(M) n 2(M) n 1+n 2. (M \mathcal{B})_{n_1} \wedge (M \mathcal{B})_{n_2} \longrightarrow (M \mathcal{B})_{n_1 + n_2} \,.

The Thom spectrum of the framing structure (exmpl.) is equivalently the sphere spectrum

M1𝕊. M 1 \simeq \mathbb{S} \,.

Because in this case B n*B_n \simeq \ast and so E n nE^{\mathcal{B}}_n \simeq \mathbb{R}^n, whence Th(E n )S nTh(E^{\mathcal{B}}_n) \simeq S^n.

For spherical fibrations



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

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


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

(Σ Ω gl 1):E Ringsgl 1Σ Ω Spec con, (\Sigma^\infty \Omega^\infty \dashv gl_1) \colon 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).


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

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



bgl 1(R)Σgl 1(R) b gl_1(R) \coloneqq \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)

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 \,.

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.


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):


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

GΩ g:BBGL 1(S) G \coloneqq \Omega^\infty g \colon 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. is equivalent to the Thom spectrum of the spherical fibration, according to def. .

This is in (ABGHR, section 8).

Equivalently the Thom spectrum is characterized as follows:


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).


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) \,.

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).


Ring spectrum structure

The universal Thom spectrum, def. , naturally inherits the structure of a ring spectrum as follows.


There are canonical commuting diagrams

(E(O(k)×O())×O(k)×O()) k EO(k+)×O(k+) k+ BO(k)×BO() BO(k+). \array{ ( E (O(k)\times O(\ell))\underset{O(k)\times O(\ell)}{\times}) \mathbb{R}^k \oplus \mathbb{R}^\ell &\longrightarrow& E O(k+\ell)\underset{O(k + \ell)}{\times} \mathbb{R}^{k+\ell} \\ \downarrow && \downarrow \\ B O(k)\times B O(\ell) &\stackrel{}{\longrightarrow}& B O(k + \ell) } \,.

Applying the Thom space functor to the top morphisms and using prop. gives morphisms

MO(k)MO()MO(k+) M O(k) \wedge M O(\ell) \longrightarrow M O(k + \ell)

that combine to a functor with smash products and hence give MOM O the structure of a ring spectrum.

More abstractly, the sufficient condition for a Thom spectrum of an ∞-module bundle (as above) to have an E-∞ ring structure is that it arises as the (∞,1)-colimit of a homomorphism of E-∞ spaces BALineB \to A Line (ABG 10, prop.6.21).

Relation to the cobordism ring


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 MOlim kπ n+kMO(k). \Omega^{un}_\bullet \simeq \pi_\bullet M O \coloneqq {\lim_{\to}}_{k \to \infty} \pi_{n+k} M O(k) \,.

This is a seminal result due to (Thom 54), whose proof proceeds by the Pontryagin-Thom construction. The presentation of the following proof follows (Francis, lecture 3).


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 we proceed 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.



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)-colimit 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.

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).


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

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory\;M(B,f) (B-bordism):

relative bordism theories:

equivariant bordism theory:

global equivariant bordism theory:


The following terms all refer to essentially the same concept:



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

Further original articles include

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

Textbook accounts include

Lecture notes include

As symmetric spectra:

As orthogonal spectra and as equivariant spectra

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)

See also

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.


  • L. Gaunce 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

Last revised on March 4, 2024 at 23:25:48. See the history of this page for a list of all contributions to it.