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Compact hyperbolic manifolds without spin structures

Bruno Martelli, Stefano Riolo and Leone Slavich

Geometry & Topology 24 (2020) 2647–2674
Abstract

We exhibit the first examples of compact, orientable, hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions n 4.

The core of the argument is the construction of a compact, oriented, hyperbolic 4–manifold M that contains a surface S of genus 3 with self-intersection 1. The 4–manifold M has an odd intersection form and is hence not spin. It is built by carefully assembling some right-angled 120–cells along a pattern inspired by the minimum trisection of 2.

The manifold M is also the first example of a compact, orientable, hyperbolic 4–manifold satisfying either of these conditions:

  • H2(M, ) is not generated by geodesically immersed surfaces.
  • There is a covering M˜ that is a nontrivial bundle over a compact surface.
Keywords
nonspin, compact, hyperbolic, manifold, $120$–cell
Mathematical Subject Classification 2010
Primary: 57M50, 57N16, 57R15
References
Publication
Received: 22 August 2019
Revised: 18 January 2020
Accepted: 19 February 2020
Published: 29 December 2020
Proposed: Ian Agol
Seconded: John Lott, Tobias H Colding
Authors
Bruno Martelli
Dipartimento di Matematica
Università di Pisa
Pisa
Italy
Stefano Riolo
Institut de mathématiques
Université de Neuchâtel
Neuchâtel
Switzerland
Leone Slavich
Dipartimento di Matematica
Università di Pisa
Pisa
Italy