---
tag: 6H8AGEN6NY
kind: concept
name: Thom spectrum
msc: [55P42, 55N22, 55R25]
specializes: [1GHF4EKBXS]
sources:
  - url: https://en.wikipedia.org/wiki/Thom_space
---

## Summary

A Thom spectrum is a [[spectrum|1GHF4EKBXS]] whose spaces are the [[Thom spaces|5MNYT3AFPH]] of a compatible sequence of vector bundles.
The basic examples are $MO$, $MSO$ and $MU$, built from the universal bundles over the classifying spaces $BO(n)$, $BSO(n)$ and $BU(n)$.
By the Pontryagin–Thom construction, the homotopy groups of these spectra are the [[cobordism|T8BVGPPSZP]] groups of manifolds with the matching structure, as in [[Thom's cobordism theorem|HM23BV561Z]].
So Thom spectra turn geometric classification problems into problems of stable homotopy theory.

## Explanations

### L1 human

The Thom space of a vector bundle is obtained by adding one point at infinity to every fibre and then gluing all these points together.
If one adds a trivial line to a bundle, its Thom space becomes the suspension of the old Thom space.
A Thom spectrum strings together Thom spaces of bigger and bigger bundles with these suspension maps.
For example, $MO$ uses the universal $n$-plane bundle over $BO(n)$ for every $n$.
Its zeroth homotopy group is $\Z/2$, which counts finite sets of points up to cobordism: a set is a boundary exactly when it has an even number of points.