On the nonexistence of elements of Kervaire invariant one

By M. Hill, M. Hopkins, D. Ravenel. 2009

523 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7b42746864012a1c7c44027dfb6ffa0acec4cd69

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The C2–spectrum Tmf1(3) and its invertible modules (2015, methodology): “A more detailed treatment can be found in [28] or [27].”
Incomplete Tambara functors (2016, methodology): “ing spectra have a coherent collection of multiplicative norm maps, studied extensively rst by Greenlees and May [4] and utilized to great eect in the work of the second author, Hopkins, and Ravenel [7] resolving the Kervaire invariant one problem. One of the most surprising observations emerging fr”
Global Homotopy Theory (2018, methodology): “Our definition of orthogonal G-spectra is not the same as the one used by Mandell and May [108] and Hill, Hopkins and Ravenel [77], who define orthogonal G-spectra as G-functors on a G-enriched extension of the 232 Equivariant stable homotopy theory category O that contains all G-representations as ”
Genuine equivariant operads (2017, background): “s. For our purposes, however, we instead focus on the perspective of Blumberg-Hill in [3], which concerns the Hill-Hopkins-Ravenel norm maps featured in the solution of the Kervaire invariant problem [15]. Given a G-spectrum Rand ﬁnite G-set Xwith nelements, the corresponding norm is a G-spectrum NX”
Uniqueness of the multiplicative cyclotomic trace (2011, methodology): “To resolve this problem, we employ the notion of ﬂat objects and functors (e.g., see [31, B.4] Recall that a functor between model categories is ﬂat if it preserves weak equivalences and colimits An object X of a model category (whose underlying category is monoidal with respect to a tensor product…”
Model structures on commutative monoids in general model categories (2014, background): “the stable equivalences, so X →φ∗φ!X is a weak equivalence in the positive (non-ﬂat) stable model structure as well. The rest of the hypotheses of Theorem 4.6 are satisﬁed, as can be seen in [34] and [21]. The rectiﬁcation axiom in the positive (non-ﬂat) stab le model structure is implied by the rec”
The spectrum of the equivariant stable homotopy category of a finite group (2015, background): “d analogues of the geometric ﬁxed point functors). In any case, the reader will ﬁnd a modern detailed discussion of the fundamental features of equivariant stable homotopy theory in the Appendices of [HHR16], while the triangulated category aspects are discussed in [HPS97]. Before we begin, let us g”
Motivic slices and coloured operads (2010, methodology): “ an axiomatic setup which allows for a wider use of ourmain results. This was prompted by the current interest in slice techniques, as exempliﬁed by the solution of the Kervaire invariant one problem [12]. Knowledge of the slices of a motivic spectrum Eprovides valuable input in Voevodsky’s slice sp”
Equivariant symmetric monoidal structures (2016, methodology): “The associated Gsymmetric monoidal structure is studied extensively in Hill-Hopkins-Ravenel.”