Closed hyperbolic manifolds without $\text{spin}^c$ structures

TL;DR

This paper proves the existence of infinitely many closed hyperbolic manifolds in dimensions five and higher that lack $ ext{spin}^c$ structures, with specific non-vanishing Stiefel--Whitney classes, expanding understanding of their topological properties.

Contribution

It demonstrates the existence of infinitely many such manifolds in all dimensions $n \,\ge\, 5$, with particular non-vanishing characteristic classes, and shows they are arithmetic of simplest type.

Findings

Existence of infinitely many hyperbolic manifolds without $ ext{spin}^c$ structures in dimensions $n \ge 5$.

Construction of manifolds with non-zero $w_{4k-1}$ classes for $n \ge 4k+1$.

All constructed manifolds are arithmetic of simplest type.

Abstract

In all dimensions $n \ge 5$, we prove the existence of closed orientable hyperbolic manifolds that do not admit any $\text{spin}^c$ structure, and in fact we show that there are infinitely many commensurability classes of such manifolds. These manifolds all have non-vanishing third Stiefel--Whitney class $w_3$ and are all arithmetic of simplest type. More generally, we show that for each $k \ge 1$ and $n \ge 4k+1$, there exist infinitely many commensurability classes of closed orientable hyperbolic $n$-manifolds $M$ with $w_{4k-1}(M) \ne 0$.