Contents

# Contents

## Idea

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

$Ker(X) \in \mathbb{Z}_2$

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

## Properties

### Nontrivial Kervaire invariants

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).

This is the statement of (the solution to) the Arf-Kervaire invariant problem.

### For surfaces and Relation to theta characteristic

On a surface $\Sigma$ a framing is equivalently a spin structure. If the surface carries a complex manifold structure then a spin structure is equivalently a theta characteristic, hence a square root $L$ of the canonical bundle. Given this, the Kervaire invariant in $\Sigma$ is equal to the dimension mod 2

$dim(H^0(\Sigma,L)) \; mod \; 2$

of the space of holomorphic sections of $L$:

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