nLab
Kervaire invariant

Idea

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

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

Related concepts

manifold dimension	invariant	quadratic form	quadratic refinement
$4k$	signature genus	intersection pairing	integral Wu structure
$4k+2$	Kervaire invariant		framing

References

• W. Browder, The Kervaire invariant of framed manifolds and its generalization, Annals of Mathematics 90 (1969), 157–186.

• John Jones, Elmer Rees, A note on the Kervaire invariant (pdf)

• Wikipedia, Kervaire invariant