# Compact hyperbolic manifolds without spin structures

## TL;DR

This paper constructs the first known examples of compact orientable hyperbolic manifolds without spin structures, demonstrating their existence in all dimensions ≥4 and exploring their topological properties.

## Contribution

It provides explicit constructions of hyperbolic manifolds lacking spin structures in all dimensions ≥4, using novel assembly techniques inspired by complex projective plane trisections.

## Key findings

- Existence of hyperbolic manifolds without spin structures in all dimensions ≥4
- Construction of a 4-manifold with an odd intersection form
- Examples where homology is not generated by geodesic surfaces or admits special bundles

## 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 orientable 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 $\mathbb{C}\mathbb{P}^2$. The manifold $M$ is also the first example of a compact orientable hyperbolic $4$-manifold satisfying any of these conditions: 1) $H_2(M,\mathbb{Z})$ is not generated by geodesically immersed surfaces. 2) There is a covering $\tilde{M}$ that is a non-trivial bundle over a compact surface.