---
tag: P4YXE08ZXF
kind: concept
name: Kervaire invariant
msc: [55Q45, 57R15]
sources:
  - url: https://en.wikipedia.org/wiki/Kervaire_invariant
same_as:
  wikidata: Q6394805
  wikipedia: Kervaire invariant
---

## Summary

The Kervaire invariant is an invariant in $\Z/2$ of a closed framed smooth manifold of dimension $4k + 2$.
It is the Arf invariant of a quadratic refinement, built from the framing, of the mod $2$ intersection form on the middle cohomology.
Michel Kervaire used it in 1960 to give a closed topological manifold of dimension $10$ with no smooth structure.
Through the Pontryagin–Thom construction it becomes a homomorphism on the [[stable homotopy groups of spheres|HGZAYPQBXW]].
The question of where it is nonzero is the Kervaire invariant problem.

## Explanations

### L1 human

A framing of a manifold is a consistent choice of a basis of directions at every point.
On a framed manifold of dimension $4k + 2$, the framing lets one assign to each middle-dimensional mod $2$ class a value $0$ or $1$.
These values form a quadratic form over the field with two elements, and such a form has a single invariant, its Arf invariant.
That invariant is the Kervaire invariant.
For example, the torus $S^1 \times S^1$ with the framing that comes from its structure as a Lie group has Kervaire invariant $1$.