Thom space



The Thom space Th(V)Th(V) of a real vector bundle VXV \to X over a topological space XX is the topological space obtained by first forming the disk bundle D(V)D(V) of (unit) disks in the fibers of VV (with respect to a metric given by any choice of orthogonal structure) and then identifying to a point the boundaries of all the disks, i.e. forming the quotient topological space by the sphere bundle S(V)S(V):

Th(V)D(V)/S(V). Th(V) \coloneqq D(V)/S(V) \,.

(N.B.: this is a quotient of the total spaces of the bundles taken in TopTop, not a bundle quotient in Top/VTop/V.)

This is equivalently the mapping cone

S(V) * p X Th(V) \array{ S(V) &\longrightarrow& * \\ {}^{\mathllap{p}}\downarrow &\swArrow& \downarrow \\ X & \longrightarrow & Th(V) }

in Top of the sphere bundle of VV. Therefore more generally, for PXP \to X any n-sphere-fiber bundle over XX (spherical fibration), its Thom space is the the mapping cone

P * p X Th(P) \array{ P &\longrightarrow& * \\ {}^{\mathllap{p}}\downarrow &\swArrow& \downarrow \\ X & \longrightarrow & Th(P) }

of the bundle projection.

For XX a compact topological space, Th(V)Th(V) is a model for the one-point compactification of the total space VV.

The Thom space of the rank-nn universal vector bundle over the classifying space BO(n)B O(n) of the orthogonal group is usuelly denoted MO(n)M O(n). As nn ranges, these spaces form the Thom spectrum.



Let XX be a topological space and let VXV \to X be a vector bundle over XX of rank nn, which is associated to an O(n)-principal bundle. Equivalently this means that VXV \to X is the pullback of the universal vector bundle E nBO(n)E_n \to B O(n) over the classifying space. Since O(n)O(n) preserves the metric on n\mathbb{R}^n, by definition, such VV inherits the structure of a metric space-fiber bundle. With respect to this structure:

  1. the unit disk bundle D(V)XD(V) \to X is the subbundle of elements of norm 1\leq 1;

  2. the unit sphere bundle S(V)XS(V)\to X is the subbundle of elements of norm =1= 1;

    S(V)i VD(V)VS(V) \overset{i_V}{\hookrightarrow} D(V) \hookrightarrow V;

  3. the Thom space Th(V)Th(V) is the cofiber (formed in Top (prop.)) of i Vi_V

    Th(V)cofib(i V) Th(V) \coloneqq cofib(i_V)

    canonically regarded as a pointed topological space.

S(V) i V D(V) (po) * Th(V). \array{ S(V) &\overset{i_V}{\longrightarrow}& D(V) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) } \,.

If VXV \to X is a general real vector bundle, then there exists an isomorphism to an O(n)O(n)-associated bundle and the Thom space of VV is, up to based homeomorphism, that of this orthogonal bundle.


If the rank of VV is positive, then S(V)S(V) is non-empty and then the Thom space is the quotient topological space

Th(V)D(V)/S(V). Th(V) \simeq D(V)/S(V) \,.

However, in the degenerate case that the rank of VV vanishes, hence the case that V=X× 0XV = X\times \mathbb{R}^0 \simeq X, then D(V)VXD(V) \simeq V \simeq X, but S(V)=S(V) = \emptyset. Hence now the pushout defining the cofiber is

i V X (po) * Th(V)X *, \array{ \emptyset &\overset{i_V}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) \simeq X_* } \,,

which exhibits Th(V)Th(V) as the coproduct of XX with the point, hence as XX with a basepoint freely adjoined.

Th(X× 0)=Th(X)X +. Th(X \times \mathbb{R}^0) = Th(X) \simeq X_+ \,.


Behaviour under direct sum of vector bundles


Let V 1,V 2XV_1,V_2 \to X be two real vector bundles. Then the Thom space (def. 1) of the direct sum of vector bundles V 1V 2XV_1 \oplus V_2 \to X is expressed in terms of the Thom space of the pullbacks V 2| D(V 1)V_2|_{D(V_1)} and V 2| S(V 1)V_2|_{S(V_1)} of V 2V_2 to the disk/sphere bundle of V 1V_1 as

Th(V 1V 2)Th(V 2| D(V 1))/Th(V 2| S(V 1)). Th(V_1 \oplus V_2) \simeq Th(V_2|_{D(V_1)})/Th(V_2|_{S(V_1)}) \,.

Notice that

  1. D(V 1V 2)D(V 2| IntD(V 1))S(V 1)D(V_1 \oplus V_2) \simeq D(V_2|_{Int D(V_1)}) \cup S(V_1);

  2. S(V 1V 2)S(V 2| IntD(V 1))IntD(V 2| S(V 1))S(V_1 \oplus V_2) \simeq S(V_2|_{Int D(V_1)}) \cup Int D(V_2|_{S(V_1)}).

(Since a point at radius rr in V 1V 2V_1 \oplus V_2 is a point of radius r 1rr_1 \leq r in V 2V_2 and a point of radius r 2r 1 2\sqrt{r^2 - r_1^2} in V 1V_1.)


For VV a vector bundle then the Thom space (def. 1) of nV\mathbb{R}^n \oplus V, the direct sum of vector bundles with the trivial rank nn vector bundle, is homeomorphic to the smash product of the Thom space of VV with the nn-sphere (the nn-fold reduced suspension).

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

Apply prop. 1 with V 1= nV_1 = \mathbb{R}^n and V 2=VV_2 = V. Since V 1V_1 is a trivial bundle, then

V 2| D(V 1)V 2×D n V_2|_{D(V_1)} \simeq V_2\times D^n

(as a bundle over X×D nX\times D^n) and similarly

V 2| S(V 1)V 2×S n. V_2|_{S(V_1)} \simeq V_2\times S^n \,.

Prop. 2 implies that for every vector bundle VV the sequence of spaces Th( nV)Th(\mathbb{R}^n \oplus V) forms a suspension spectrum: this is the Thom spectrum of VV.


By prop. 2 and remark 1 the Thom space (def. 1) of a trivial vector bundle of rank nn is the nn-fold suspension of the base space

Th(X× n) S nTh(X× 0) S n(X +). \begin{aligned} Th(X \times \mathbb{R}^n) & \simeq S^n \wedge Th(X\times \mathbb{R}^0) \\ & \simeq S^n \wedge (X_+) \end{aligned} \,.

Therefore a general Thom space may be thought of as “twisted suspension”, with twist encoded by a vector bundle (or rather by its underlying spherical fibration). See at Thom spectrum – For infinity-module bundles for more on this.


For V 1X 1V_1 \to X_1 and V 2X 2V_2 \to X_2 to 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. 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. 3 induces on the Thom spectra of remark 2 the structure of ring spectra.


The Thom isomorphism for Thom spaces was originally found in

  • René Thom, Quelques propriétés globales des variétés différentiables Comm. Math. Helv. , 28 (1954) pp. 17–86

For general discussion see


  • Robert Stong, Notes on cobordism theory , Princeton Univ. Press (1968)

  • W.B. Browder, Surgery on simply-connected manifolds , Springer (1972)

Revised on May 19, 2016 05:52:03 by Urs Schreiber (