the Thom space
/ TOM /
Given a vector bundle E over X, the total space is non-compact: every fiber is an infinite R^n stretching to infinity. The Thom space is a way to tidy this up by collapsing all those infinities into a single point, producing a compact pointed space that packages the bundle's twisting into honest homotopy data. It is the bundle's natural 'one-point compactification fiberwise', and it is the bridge from bundles to stable homotopy.
Concretely, put a metric on E and take the unit disk bundle D(E) (vectors of length at most 1) and the unit sphere bundle S(E) (vectors of length exactly 1). The Thom space is the quotient Th(E) = D(E) / S(E): you crush the boundary sphere of every fiber to one common basepoint. Equivalently it is the one-point compactification of the total space E. When X is a point, E is just R^n and Th(E) is the n-sphere S^n; for a general bundle of rank n, Th(E) looks like a 'twisted suspension' of X, an S^n-bundle's worth of spheres all sharing the point at infinity.
Thom spaces are the heart of several constructions: the Thom isomorphism relates the cohomology of X to that of Th(E) shifted by the rank, the Thom class lives in the cohomology of Th(E), and cobordism theory is built from Thom spaces of universal bundles (the Thom spectrum MO, MU). They also give the cleanest route to the Euler class and to Pontryagin-Thom collapse, which converts submanifolds into maps to Thom spaces. The subtlety to respect: the Thom space depends on the bundle, not just on X — different bundles over the same base give genuinely different Thom spaces, which is the whole point of using them to detect twisting.
For the trivial rank-n bundle X times R^n, the Thom space is the n-fold suspension of X with a disjoint basepoint added: Th(trivial) = Sigma^n (X_+). In particular over a point the trivial R^n gives Th = S^n, and over X the trivial line bundle gives the (unreduced) suspension of X. Twisting the bundle changes this suspension into a non-product 'twisted' version.
Trivial bundles give plain suspensions; the twist in a nontrivial bundle deforms the suspension into something new.
The Thom space depends on the actual bundle, not merely on the base or even the bundle's rank — that dependence is the source of its power. Treating Th(E) as if it only remembered X (or only the rank) discards exactly the twisting information it was built to record.