# nLab Thom space

### Context

#### Topology

topology

algebraic topology

# Contents

## Idea

The Thom space $Th(V)$ of a vector bundle $V \to X$ over a topological space $X$ is the topological 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 $S(V)$:

$Th(V) := D(V)/S(V) \,.$

This is equivalently the mapping cone

$\array{ S(V) &\to& * \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\to& Th(V) }$

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

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

of the bundle projection.

For $X$ a compact topological space $Th(V)$ is the one-point compactification of the total space $V$.

## Properties

###### Observation

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

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

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

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

is the smash product of the Thom space of $V$ with the $n$-sphere (the $n$-fold suspension).

###### Example

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

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

is the smash product of the $n$-sphere with $X$ with one base point freely adjoined (the $n$-fold suspension).

###### Remark

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

## 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

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

Also

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

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

Revised on May 27, 2014 11:19:08 by Benjamin Antieau (24.22.243.35)