For $X$ a framed smooth manifold of dimension $4k +2$, $k \in \mathbb{N}$, 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.
Manifolds with non-trivial Kervaire invariant, hence with Kervaire invariant 1, exist in dimension
$d = 2 = 4\cdot 0 + 2$
$d = 6 = 4\cdot 1 + 2$
$d = 14 = 4 \cdot 3 + 2$
$d = 30 = 4 \cdot 7 + 2$
$d = 62 = 4 \cdot 15 + 2$
and in no other dimension, except possibly in $d = 126$ (a case that is still open).
manifold dimension | invariant | quadratic form | quadratic refinement |
---|---|---|---|
$4k$ | signature genus | intersection pairing | integral Wu structure |
$4k+2$ | 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:
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