nLab Kervaire invariant

Context

Manifolds and cobordisms

manifolds and cobordisms

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.

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