On the non-existence of elements of Kervaire invariant one
Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel
TL;DR
This paper proves that elements with Kervaire invariant one in the homotopy groups of spheres only exist up to dimension 126, resolving a major open problem in algebraic topology except for the case of dimension 126.
Contribution
It establishes the non-existence of Kervaire invariant one elements in all dimensions greater than 126, confirming the limited dimensions where such elements can occur.
Findings
Kervaire invariant one elements exist only in dimensions up to 126
Smooth framed manifolds with Kervaire invariant one are limited to specific dimensions
The longstanding problem in algebraic topology is resolved except for the dimension 126 case
Abstract
We show that Kervaire invariant one elements in the homotopy groups of spheres exist only in dimensions at most 126. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. With the exception of dimension 126 this resolves a longstanding problem in algebraic topology.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology · Algebraic Geometry and Number Theory