nLab Thom space

Context

Topology

topology

algebraic topology

Contents

Idea

The Thom space $\mathrm{Th}\left(V\right)$ of a vector bundle $V\to X$ over a topological space $X$ is the space obtained by first forming the disk bundle $D\left(V\right)$ 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 $S\left(V\right)$:

$\mathrm{Th}\left(V\right):=D\left(V\right)/S\left(V\right)\phantom{\rule{thinmathspace}{0ex}}.$Th(V) := D(V)/S(V) \,.

This is equivalently the mapping cone

$\begin{array}{ccc}S\left(V\right)& \to & *\\ {↓}^{p}& & ↓\\ X& \to & \mathrm{Th}\left(V\right)\end{array}$\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

$\begin{array}{ccc}P& \to & *\\ {↓}^{p}& & ↓\\ X& \to & \mathrm{Th}\left(P\right)\end{array}$\array{ P &\to& * \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\to& Th(P) }

of the bundle projection.

Properties

Observation

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

$\mathrm{Th}\left(X×{ℝ}^{0}\right)=\mathrm{Th}\left(X\right)\simeq {X}_{+}$Th(X \times \mathbb{R}^0) = Th(X) \simeq X_+
Proposition

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

$\mathrm{Th}\left(V\oplus {ℝ}^{n}\right)\simeq {S}^{n}\wedge \mathrm{Th}\left(V\right)$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 ${ℝ}^{n}×X\to X$ is a trivial vector bundle of rank $n$, then

$\mathrm{Th}\left(X×{ℝ}^{n}\right)\simeq {S}^{n}\wedge {X}_{+}$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 $\mathrm{Th}\left({ℝ}^{n}\oplus V\right)$ forms an Omega-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 31, 2011 11:05:42 by Urs Schreiber (131.211.238.226)