CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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 $S(V)$:
This is equivalently the mapping cone
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
of the bundle projection.
For $X$ a compact topological space $Th(V)$ is the one-point compactification of the total space $V$.
The Thom space of the rank-0 bundle over $X$ is the space $X$ with a basepoint freely adjoined:
For $V$ a vector bundle and $\mathbb{R}^n \oplus V$ its fiberwise direct sum with the trivial rank $n$ vector bundle we have
is the smash product of the Thom space of $V$ with the $n$-sphere (the $n$-fold suspension).
In particular, if $\mathbb{R}^n \times X \to X$ is a trivial vector bundle of rank $n$, then
is the smash product of the $n$-sphere with $X$ with one base point freely adjoined (the $n$-fold suspension).
This implies that for every vector bundle $V$ the sequence of spaces $Th(\mathbb{R}^n \oplus V)$ forms an Omega-spectrum: this is called the Thom spectrum of $V$.
The Thom isomorphism for Thom spaces was originally found in
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)
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)