Compact hyperbolic manifolds without spin structures

Martelli, Bruno; Riolo, Stefano; Slavich, Leone

doi:10.2140/gt.2020.24.2647

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Compact hyperbolic manifolds without spin structures

Bruno Martelli, Stefano Riolo and Leone Slavich

Geometry & Topology 24 (2020) 2647–2674

DOI: 10.2140/gt.2020.24.2647

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 $n \geq 4$ .

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 ${ℂ ℙ}^{2}$ .

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

$H_{2} (M, ℤ)$ is not generated by geodesically immersed surfaces.
There is a covering $\tilde{M}$ 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

Volume 24, issue 5 (2020)