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On
the high-dimensional geography problem
Robert Burklund and Andrew Senger |
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Geometry & Topology 28 (2024) 4257–4293
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Abstract |
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In 1962, Wall showed that smooth, closed, oriented, –connected –manifolds of dimension at least are classified up to connected sum with an exotic sphere by an algebraic refinement of the intersection form, which he called an –space. We complete the determination of which –spaces are realizable by smooth, closed, oriented, – connected –manifolds for all . In dimension , 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 , where we provide a counterexample. This counterexample is related to the Witten genus and its refinement to a map of –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. |
Keywords
highly connected manifold, exotic sphere, synthetic
spectra, Adams spectral sequence, $J$–homomorphism,
topological modular forms, Witten genus
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Mathematical Subject Classification
Primary: 55N34, 55T15, 57R19, 57R60
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References |
Publication
Received: 17 March 2022
Revised: 25 June 2023
Accepted: 30 July 2023
Published: 27 December 2024
Proposed: Mark Behrens
Seconded: Benson Farb, Nathalie Wahl
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Authors |
| Robert Burklund | |
| Department of Mathematical
Sciences University of Copenhagen Copenhagen Denmark |
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| Andrew Senger | |
| Department of Mathematics Harvard University Cambridge, MA United States |
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| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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