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_+ \,.


Homotopy-theoretic nature


Let VXV \to X be a vector bundle over a CW-complex XX. Then the Thom space Th(V)Th(V) (def. 1) is equivalently the homotopy cofiber (def.) of the inclusion S(V)D(V)S(V) \longrightarrow D(V) of the sphere bundle into the disk bundle.


The Thom space is defined as the ordinary cofiber of S(V)D(V)S(V)\to D(V). Under the given assumption, this inclusion is a relative cell complex inclusion, hence a cofibration in the classical model structure on topological spaces (thm.). Therefore in this case the ordinary cofiber represents the homotopy cofiber (def.).

The equivalence to the following alternative model for this homotopy cofiber is relevant when discussing Thom isomorphisms and orientation in generalized cohomology:


Let VXV \to X be a vector bundle over a CW-complex XX. Write VXV-X for the complement of its 0-section. Then the Thom space Th(V)Th(V) (def. 1) is homotopy equivalent to the mapping cone of the inclusion (VX)V(V-X) \hookrightarrow V (hence to the pair (V,VX)(V,V-X) in the language of generalized (Eilenberg-Steenrod) cohomology).


The mapping cone of any map out of a CW-complex represents the homotopy cofiber of that map (exmpl.). Moreover, transformation by (weak) homotopy equivalences between morphisms induces a (weak) homotopy equivalence on their homotopy fibers (prop.). But we have such a weak homotopy equivalence, given by contracting away the fibers of the vector bundle:

VX V W cl W cl S(V) D(V). \array{ V-X &\longrightarrow& V \\ {}^{\mathllap{\in W_{cl}}}\downarrow && \downarrow^{\mathrlap{\in W_{cl}}} \\ S(V) &\hookrightarrow& D(V) } \,.

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. 3 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. 4 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. 4 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 a “twisted reduced 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.

Correspondingly the Thom isomorphism for a given Thom space is a twisted version of the suspension isomorphism.


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. 5 induces on the Thom spectra of remark 2 the structure of ring spectra.


If the base space of the vector bundle carries the structure of a CW-complex, then its Thom space (def. 1) canonically inherits the structure of a CW-complex, too:


Let VXV \to X be a vector bundle of rank n1n \geq 1. over a CW-complex XX.

Then Th(V)Th(V) has the structure of a CW-complex with

  1. S(E)/S(E)S(E)/S(E) the only 0-cell

  2. precisely one (n+k)(n+k)-cell D k+nTh(V)D^{k+n}\to Th(V) for each kk-cell D kXD^k \to X of XX, given as the pullback

    D k+n D(V) D(V)/S(V)=Th(V) (pb) D k X. \array{ D^{k+n} &\longrightarrow& D(V) &\longrightarrow& D(V)/S(V) = Th(V) \\ \downarrow &(pb)& \downarrow \\ D^k &\longrightarrow& X } \,.

(e.g. Cruz 04, lemma 6)

In particular, Th(V)Th(V) has a single nn-cell and an (n+1)(n+1)-cell for each 1-cell of XX. There are no cells in Th(C)Th(C) between dimension 00 and nn. The cellular boundary of an (n+1)(n+1)-cell is 0 if VV is orientable over the corresponding 1-cell of XX, and it is twice the nn-cell in the opposite case. Thus H n(Th(V);)H^n(Th(V);\mathbb{Z}) is \mathbb Z if V\mathbb ZV is orientable and 00 if VV is non-orientable. In the orientable case a generator of H^n(Th(V);{\mathbb Z}) restricts to a generator of H n(S n;)H^n(Th(V);{\mathbb ZH^n(S^n;\mathbb{Z}) in the “fiber” S nS^n of Th(V)Th(V) over the 0-cell of XX, hence the same is true for all the “fibers” S nS^n and so one has a Thom class.

(MO discussion)



Given a vector bundle VXV \to X of rank nn, then the reduced ordinary cohomology of its Thom space Th(V)Th(V) (def. 1) vanishes in degrees <n\lt n:

H˜ <n(Th(V))H <n(D(V),S(V))0. \tilde H^{\bullet \lt n}(Th(V)) \simeq H^{\bullet \lt n}(D(V), S(V)) \simeq 0 \,.

Consider the long exact sequence of relative cohomology (here)

H 1(D(V))i *H 1(S(V))H (D(V),S(V))H (D(V))i *H (S(V)). \cdots \to H^{\bullet-1}(D(V)) \overset{i^\ast}{\longrightarrow} H^{\bullet-1}(S(V)) \longrightarrow H^\bullet(D(V), S(V)) \longrightarrow H^{\bullet}(D(V)) \overset{i^\ast}{\longrightarrow} H^{\bullet}(S(V)) \to \cdots \,.

Since the cohomology in degree kk only depends on the kk-skeleton, and since for k<nk \lt n the kk-skeleton of S(V)S(V) equals that of XX, and since D(V)D(V) is even homotopy equivalent to XX, the morhism i *i^\ast is an isomorphism in degrees lower than nn. Hence by exactness of the sequence it follows that H <n(D(V),S(V))=0H^{\bullet \lt n}(D(V),S(V)) = 0.


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

See also

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

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

  • Martin Vito Cruz, An introduction to cobordism, 2004 (pdf)

Revised on July 7, 2016 05:03:54 by Urs Schreiber (