# nLab Thom space

bundles

## Examples and Applications

#### Topology

topology

algebraic topology

# Contents

## Idea

The Thom space $Th(V)$ of a real 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 $V$ (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 $S(V)$:

$Th(V) \coloneqq D(V)/S(V) \,.$

(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

$\array{ S(V) &\longrightarrow& * \\ {}^{\mathllap{p}}\downarrow &\swArrow& \downarrow \\ X & \longrightarrow & Th(V) }$

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

$\array{ P &\longrightarrow& * \\ {}^{\mathllap{p}}\downarrow &\swArrow& \downarrow \\ X & \longrightarrow & Th(P) }$

of the bundle projection.

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

The Thom space of the rank-$n$ universal 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.

## Definition

###### Definition

Let $X$ be a topological space and let $V \to X$ be a vector bundle over $X$ of rank $n$, which is associated to an O(n)-principal bundle. Equivalently this means that $V \to X$ is the pullback of the universal vector bundle $E_n \to B O(n)$ over the classifying space. Since $O(n)$ preserves the metric on $\mathbb{R}^n$, by definition, such $V$ inherits the structure of a metric space-fiber bundle. With respect to this structure:

1. the unit disk bundle $D(V) \to X$ is the subbundle of elements of norm $\leq 1$;

2. the unit sphere bundle $S(V)\to X$ is the subbundle of elements of norm $= 1$;

$S(V) \overset{i_V}{\hookrightarrow} D(V) \hookrightarrow V$;

3. the Thom space $Th(V)$ is the cofiber (formed in Top (prop.)) of $i_V$

$Th(V) \coloneqq cofib(i_V)$

canonically regarded as a pointed topological space.

$\array{ S(V) &\overset{i_V}{\longrightarrow}& D(V) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) } \,.$

If $V \to X$ is a general real vector bundle, then there exists an isomorphism to an $O(n)$-associated bundle and the Thom space of $V$ is, up to based homeomorphism, that of this orthogonal bundle.

###### Remark

If the rank of $V$ is positive, then $S(V)$ is non-empty and then the Thom space is the quotient topological space

$Th(V) \simeq D(V)/S(V) \,.$

However, in the degenerate case that the rank of $V$ vanishes, hence the case that $V = X\times \mathbb{R}^0 \simeq X$, then $D(V) \simeq V \simeq X$, but $S(V) = \emptyset$. Hence now the pushout defining the cofiber is

$\array{ \emptyset &\overset{i_V}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Th(V) \simeq X_* } \,,$

which exhibits $Th(V)$ as the coproduct of $X$ with the point, hence as $X$ with a basepoint freely adjoined.

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

## Properties

### Behaviour under direct sum of vector bundles

###### Proposition

Let $V_1,V_2 \to X$ be two real vector bundles. Then the Thom space (def. 1) of the direct sum of vector bundles $V_1 \oplus V_2 \to X$ is expressed in terms of the Thom space of the pullbacks $V_2|_{D(V_1)}$ and $V_2|_{S(V_1)}$ of $V_2$ to the disk/sphere bundle of $V_1$ as

$Th(V_1 \oplus V_2) \simeq Th(V_2|_{D(V_1)})/Th(V_2|_{S(V_1)}) \,.$
###### Proof

Notice that

1. $D(V_1 \oplus V_2) \simeq D(V_2|_{Int D(V_1)}) \cup S(V_1)$;

2. $S(V_1 \oplus V_2) \simeq S(V_2|_{Int D(V_1)}) \cup Int D(V_2|_{S(V_1)})$.

(Since a point at radius $r$ in $V_1 \oplus V_2$ is a point of radius $r_1 \leq r$ in $V_2$ and a point of radius $\sqrt{r^2 - r_1^2}$ in $V_1$.)

###### Proposition

For $V$ a vector bundle then the Thom space (def. 1) of $\mathbb{R}^n \oplus V$, the direct sum of vector bundles with the trivial rank $n$ vector bundle, is homeomorphic to the smash product of the Thom space of $V$ with the $n$-sphere (the $n$-fold reduced suspension).

$Th(\mathbb{R}^n \oplus V) \simeq S^n \wedge Th(V) = \Sigma^n Th(V) \,.$
###### Proof

Apply prop. 1 with $V_1 = \mathbb{R}^n$ and $V_2 = V$. Since $V_1$ is a trivial bundle, then

$V_2|_{D(V_1)} \simeq V_2\times D^n$

(as a bundle over $X\times D^n$) and similarly

$V_2|_{S(V_1)} \simeq V_2\times S^n \,.$
###### Remark

Prop. 2 implies that for every vector bundle $V$ the sequence of spaces $Th(\mathbb{R}^n \oplus V)$ forms a suspension spectrum: this is the Thom spectrum of $V$.

###### Example

By prop. 2 and remark 1 the Thom space (def. 1) of a trivial vector bundle of rank $n$ is the $n$-fold suspension of the base space

\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 “twisted 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.

###### Proposition

For $V_1 \to X_1$ and $V_2 \to X_2$ to vector bundles, let $V_1 \boxtimes V_2 \to X_1 \times X_2$ be the direct sum of vector bundles of their pullbacks to $X_1 \times X_2$. The corresponding Thom space is the smash product of the individual Thom spaces:

$Th(V_1 \boxtimes V_2) \simeq Th(V_1) \wedge Th(V_2) \,.$
###### Remark

Prop. 3 induces on the Thom spectra of remark 2 the structure of ring spectra.

## 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 May 19, 2016 05:52:03 by Urs Schreiber (131.220.184.222)