Geometric approach to stable homotopy groups of spheres II. The Kervaire invariant

Details Geometric approach to stable homotopy groups of spheres II. The Kervaire invariant Geometric approach to stable homotopy groups of spheres II. The Kervaire invariant

2010-11-26

arxiv.org/abs/1011.5717v3

Mathematics

Algebraic Topology

Scientific paper

A solution to the Kervaire invariant problem is presented. We introduce the concepts of abelian structure on skew-framed immersions, bicyclic structure on $\Z/2^{[3]}$--framed immersions, and quaternionic-cyclic structure on $\Z/2^{[4]}$--framed immersions. Using these concepts, we prove that for sufficiently large $n$, $n=2^{\ell}-2$, an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant. Additionally, for $\ell \ge 12$ (i.e., for $n \ge 4094$) an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant if this skew-framed immersion admits a compression of order 16.

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