Year: 2024
Type: article
Abstract: In 1962, Wall showed that smooth, closed, oriented, .n1/-connected 2n-manifolds of dimension at least 6 are classified up to connected sum with an exotic sphere by an algebraic refinement of the intersection form, which he called an n-space.We complete the determination of which n-spaces are realizable by smooth, closed, oriented, .n1/connected 2n-manifolds for all n ¤ 63.In dimension 126, the Kervaire invariant one problem remains open.Along the way, we completely resolve conjectures of Galatius and Randal-Williams and Bowden, Crowley and Stipsicz, showing that they are true outside of the exceptional dimension 23, where we provide a counterexample.This counterexample is related to the Witten genus and its refinement to a map of E 1 -ring spectra by Ando, Hopkins and Rezk.By previous work of many authors, including Wall, Schultz, Stolz, and Hill, Hopkins and Ravenel, as well as recent joint work of Hahn with the authors, these questions have been resolved for all but finitely many dimensions, and the contribution of this paper is to fill in these gaps.55N34, 55T15, 57R19, 57R60 1. Introduction 4258 2. Classification of .n1/-connected 2n-manifolds 4263 3. The Witten genus of 11-connected 24-manifolds 4269 4. Reduction to homotopy theory 4273 5.A homotopy argument 4277 6.The case of dimension 17 4279 7. The case of dimension 23 at the prime 3 4281 8.The case of dimension 127 at the prime 2
Source: Geometry & Topology
Authors: Robert Burklund, Andrew Senger
Institutions: University of Copenhagen, Harvard University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Funder: National Science Foundation
OpenAlex ID: W3040909734