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

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

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

• $d = 2 = 4\cdot 0 + 2$

• $d = 6 = 4\cdot 1 + 2$

• $d = 14 = 4 \cdot 3 + 2$

• $d = 30 = 4 \cdot 7 + 2$

• $d = 62 = 4 \cdot 15 + 2$

• $d = 126 = 4 \cdot 31 + 2$

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 $L$:

Related concepts

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

• Hopf invariant, Hopf invariant one

• Arf-Kervaire invariant problem

• self-dual higher gauge theory

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

• Akhil Mathew, The Kervaire invariant I

On the solution of the Arf-Kervaire invariant problem away from $d = 126$:

• Michael Hill, Michael Hopkins, Douglas Ravenel: On the non-existence of elements of Kervaire invariant one, Annals of Mathematics 184 1 (2016) 1-262 [arXiv:0908.3724, doi:10.4007/annals.2016.184.1.1]

On the equivariant homotopy theory involved:

• Michael Hill, Michael Hopkins, Douglas Ravenel, The Arf-Kervaire problem in algebraic topology: Sketch of the proof (pdf)

More resources are collected at:

• Douglas Ravenel, A solution to the Arf-Kervaire invariant problem, web esources 2009

Solution to the last remaining case, $d = 126$, is claimed in:

• Weinan Lin, Guozhen Wang, Zhouli Xu: On the Last Kervaire Invariant Problem [arXiv:2412.10879]