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Kervaire invariant

Contents

Idea

For XX a framed smooth manifold of dimension 4k+24k +2, kk \in \mathbb{N}, the Kervaire invariant or Arf-Kervaire invariant

Ker(X) 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 (the intersection pairing).

Properties

Nontrivial Kervaire invariants

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

  • d=2=40+2d = 2 = 4\cdot 0 + 2

  • d=6=41+2d = 6 = 4\cdot 1 + 2

  • d=14=43+2d = 14 = 4 \cdot 3 + 2

  • d=30=47+2d = 30 = 4 \cdot 7 + 2

  • d=62=415+2d = 62 = 4 \cdot 15 + 2

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

For surfaces and Relation to theta characteristic

On a surface Σ\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 LL of the canonical bundle. Given this, the Kervaire invariant in Σ\Sigma is equal to the dimension mod 2

dim(H 0(Σ,L))mod2 dim(H^0(\Sigma,L)) \; mod \; 2

of the space of holomorphic sections of LL:

manifold dimensioninvariantquadratic formquadratic refinement
4k4ksignature genusintersection pairingintegral Wu structure
4k+24k+2Kervaire invariantframing

References

On the solution of the Arf-Kervaire invariant problem:

On the equivariant homotopy theory involved:

More resources are collected at

Revised on September 9, 2014 16:59:57 by Urs Schreiber (185.26.182.34)