#### Volume 24, issue 5 (2020)

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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 $n\ge 4$.

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 ${ℂℙ}^{2}$.

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

• ${H}_{2}\left(M,ℤ\right)$ is not generated by geodesically immersed surfaces.
• There is a covering $\stackrel{˜}{M}$ that is a nontrivial bundle over a compact surface.

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##### Keywords
nonspin, compact, hyperbolic, manifold, $120$–cell
##### Mathematical Subject Classification 2010
Primary: 57M50, 57N16, 57R15
##### Publication
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