nLab
Thom space

Contents

Idea

The Thom space Th(V)Th(V) of a 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 and then identifying to a point the boundaries of all the disks, i.e. forming the quotient by the sphere bundle S(V)S(V):

Th(V):=D(V)/S(V). Th(V) := 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) &\to& * \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\to& Th(V) }

in Top of the sphere bundle of VV. Therefore more generally, for PXP \to X any S nS^n-bundle over XX, its Thom space is the the mapping cone

P * p X Th(P) \array{ P &\to& * \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\to& Th(P) }

of the bundle projection.

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

The Thom space of the universal rank-nn 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.

Properties

Observation

The Thom space of the rank-0 bundle over XX is the space XX with a basepoint freely adjoined:

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

For VV a vector bundle and nV\mathbb{R}^n \oplus V its fiberwise direct sum with the trivial rank nn vector bundle we have

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

is the smash product of the Thom space of VV with the nn-sphere (the nn-fold suspension).

Example

In particular, if n×XX\mathbb{R}^n \times X \to X is a trivial vector bundle of rank nn, then

Th(X× n)S nX + Th(X \times \mathbb{R}^n) \simeq S^n \wedge X_+

is the smash product of the nn-sphere with XX with one base point freely adjoined (the nn-fold suspension).

Remark

This 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 called the Thom spectrum of VV.

References

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

Also

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

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

Revised on February 10, 2016 06:08:29 by Todd Trimble (67.81.95.215)