# 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 to a point the boundaries of all the disks, i.e. forming the quotient by the sphere bundle $S(V)$:

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

(N.B.: this is a quotient of the total spaces of the bundles taken in $Top$, not a bundle quotient in $Top/V$.) 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, the $Th(V)$ is a model for the one-point compactification of the total space $V$.

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

## 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

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)