We show that the Kervaire invariant one elements θj ∈ π2j+1−2S exist only for j ≤ 6. 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. Except for dimension 126 this resolves a longstanding problem in algebraic topology.
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Hill et al. (2016) studied this question.
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