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).
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|
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