nLab
Thom space

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 $S(V)$:

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

This is equivalently the mapping cone

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

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

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

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

$$\mathrm{Th}(X\times {ℝ}^n)\simeq S^n\wedge X_{+}$$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 $\mathrm{Th}({ℝ}^n\oplus V)$ forms an Omega-spectrum: this is called the Thom spectrum of $V$.

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)