manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
For a framed smooth manifold of dimension , , the Kervaire invariant or Arf-Kervaire invariant
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).
Manifolds with non-trivial Kervaire invariant, hence with Kervaire invariant 1, exist in dimension
and in no other dimension, except possibly in (a case that is still open).
This is the statement of (the solution to) the Arf-Kervaire invariant problem.
On a surface 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 of the canonical bundle. Given this, the Kervaire invariant in is equal to the dimension mod 2
of the space of holomorphic sections of :
manifold dimension | invariant | quadratic form | quadratic refinement |
---|---|---|---|
signature genus | intersection pairing | integral Wu structure | |
Kervaire invariant | framing |
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
On the solution of the Arf-Kervaire invariant problem:
On the equivariant homotopy theory involved:
More resources are collected at
Last revised on January 20, 2016 at 10:40:31. See the history of this page for a list of all contributions to it.