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2020 Compact hyperbolic manifolds without spin structures
Bruno Martelli, Stefano Riolo, Leone Slavich
Geom. Topol. 24(5): 2647-2674 (2020). 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 n4.

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.

Citation

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Bruno Martelli. Stefano Riolo. Leone Slavich. "Compact hyperbolic manifolds without spin structures." Geom. Topol. 24 (5) 2647 - 2674, 2020. https://doi.org/10.2140/gt.2020.24.2647

Information

Received: 22 August 2019; Revised: 18 January 2020; Accepted: 19 February 2020; Published: 2020
First available in Project Euclid: 5 January 2021

MathSciNet: MR4194300
Digital Object Identifier: 10.2140/gt.2020.24.2647

Subjects:
Primary: 57M50, 57N16, 57R15

Rights: Copyright © 2020 Mathematical Sciences Publishers

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