# nLab Kervaire invariant

### Context

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Idea

For $X$ a framed smooth manifold of dimension $4k+2$, $k\in ℕ$, the Kervaire invariant or Arf-Kervaire invariant

$\mathrm{Ker}\left(X\right)\in {ℤ}_{2}$Ker(X) \in \mathbb{Z}_2

with values in the group of order 2 is the Arf invariant? of the skew-quadratic form on the middle dimensional homology group.

## Properties

Manifolds with non-trivial Kervaire invariant, hence with Kervaire invariant 1, exist in dimension

• $d=2=4\cdot 0+2$

• $d=6=4\cdot 1+2$

• $d=14=4\cdot 3+2$

• $d=30=4\cdot 7+2$

• $d=62=4\cdot 15+2$

and in no other dimension, except possibly in $d=126$ (a case that is still open).

$4k$signature genusintersection pairingintegral Wu structure
$4k+2$Kervaire invariantframing