In mathematics, the Kervaire invariant is an invariant of a framed ( 4 k + 2 ) {\displaystyle (4k+2)} -dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf. The Kervaire invariant is defined as the Arf invariant of the skew-quadratic form on the middle dimensional homology group. It can be thought of as the simply-connected quadratic L-group L 4 k + 2 {\displaystyle L_{4k+2}} , and thus analogous to the other invariants from L-theory: the signature, a 4 k {\displaystyle 4k} -dimensional invariant (either symmetric or quadratic, L 4 k ≅ L 4 k {\displaystyle L^{4k}\cong L_{4k}} ), and the De Rham invariant, a ( 4 k + 1 ) {\displaystyle (4k+1)} -dimensional symmetric invariant L 4 k + 1 {\displaystyle L^{4k+1}} . In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1. The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero. For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. In 2024 a preprint by Weinan Lin, Guozhen Wang and Zhouli Xu, settled the case in dimension 126, proving the existence of smooth framed manifolds with Kervaire invariant one.
Definition The Kervaire invariant is the Arf invariant of the quadratic form determined by the framing on the middle-dimensional Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } -coefficient homology group
q : H 2 m + 1 ( M ; Z / 2 Z ) → Z / 2 Z , {\displaystyle q\colon H_{2m+1}(M;\mathbb {Z} /2\mathbb {Z} )\to \mathbb {Z} /2\mathbb {Z} ,}
and is thus sometimes called the Arf–Kervaire invariant. The quadratic form (properly, skew-quadratic form) is a quadratic refinement of the usual ε-symmetric form on the middle dimensional homology of an (unframed) even-dimensional manifold; the framing yields the quadratic refinement. The quadratic form q can be defined by algebraic topology using functional Steenrod squares, and geometrically via the self-intersections of immersions S 2 m + 1 {\displaystyle S^{2m+1}}
→ {\displaystyle \to } M 4 m + 2 {\displaystyle M^{4m+2}} determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings S 2 m + 1 {\displaystyle S^{2m+1}}
→ {\displaystyle \to } M 4 m + 2 {\displaystyle M^{4m+2}} (for m ≠ 0 , 1 , 3 {\displaystyle m\neq 0,1,3} ) and the mod 2 Hopf invariant of maps S 4 m + 2 + k → S 2 m + 1 + k {\displaystyle S^{4m+2+k}\to S^{2m+1+k}}
(for m = 0 , 1 , 3 {\displaystyle m=0,1,3} ).
… excerpt ends here. Continue reading the full article.
