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|
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