Contents

Idea

The Thom space $\mathrm{Th}(V)$ of a vector bundle $V\to X$ over a topological space $X$ is the space obtained by first forming the disk bundle $D(V)$ of (unit) disks in the fibers of $V$ and then identifying the boundary of each disk, i.e. forming the quotient by the sphere bundle $S(V)$:

This is equivalently the mapping cone

in Top of the sphere bundle of $V$. Therefore more generally, for $P\to X$ any $S^n$-bundle over $X$, its Thom space is the the mapping cone

of the bundle projection.

Properties

Related concepts

• one-point compactification

• Thom spectrum

• Thom isomorphism

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

• Michael Atiyah, Thom complexes, Proc. London Math. Soc. 11 (1961) pp. 291–310

• Yuli B. Rudyak?, On Thom spectra, orientability, and cobordism, Springer 1998 googB

• eom, Thom space

• Dale Husemöller, Fibre bundles , McGraw-Hill (1966)

• myyn.org Thom space, Thom class, Thom isomorphism theorem

Also

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

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