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Compact hyperbolic manifolds without spin structures

Bruno Martelli, Stefano Riolo and Leone Slavich

Geometry & Topology 24 (2020) 2647–2674

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.

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nonspin, compact, hyperbolic, manifold, $120$–cell
Mathematical Subject Classification 2010
Primary: 57M50, 57N16, 57R15
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
Bruno Martelli
Dipartimento di Matematica
Università di Pisa
Stefano Riolo
Institut de mathématiques
Université de Neuchâtel
Leone Slavich
Dipartimento di Matematica
Università di Pisa