---
tag: 5MNYT3AFPH
kind: concept
name: Thom space
msc: [55R25, 57R20]
sources:
  - url: https://en.wikipedia.org/wiki/Thom_space
same_as:
  wikidata: Q2421733
  wikipedia: Thom space
---

## Summary

The Thom space of a real vector bundle $E \to B$ is its unit disc bundle with the unit [[sphere bundle|3VQ2V5HH3R]] collapsed to a point.
For compact $B$ it is the [[one-point compactification|0GWN3CMRK8]] of the total space $E$.
The [[Thom isomorphism|N685RXHDZX]] identifies its reduced cohomology with the cohomology of $B$ shifted by the rank.
Thom spaces of universal bundles form the [[Thom spectra|6H8AGEN6NY]] that compute cobordism groups.

## Explanations

### L1 human

Over each point of the base there is a disc, the vectors of length at most one.
The Thom space glues the boundary circles or spheres of all these discs to one common point.
So each disc becomes a sphere, and all these spheres touch at the collapsed point.
For a bundle over a single point, the Thom space is one disc with its boundary collapsed, which is a sphere of the same dimension.