Hyperbolic manifolds without $\text{spin}^\mathbb{C}$ structures and non-vanishing higher order Stiefel-Whitney classes

TL;DR

This paper demonstrates the existence of hyperbolic manifolds with non-vanishing higher order Stiefel-Whitney classes and no $ ext{spin}^ ext{C}$ structures within each commensurability class of certain arithmetic hyperbolic manifolds.

Contribution

It establishes the presence of manifolds with specific topological properties, namely non-vanishing Stiefel-Whitney classes and absence of $ ext{spin}^ ext{C}$ structures, in all relevant commensurability classes.

Findings

Existence of manifolds with non-vanishing $w_{2j}$ classes for all $0 \\leq 2j \\leq n$

Existence of manifolds without $ ext{spin}^ ext{C}$ structures in each class

Results hold for all commensurability classes of certain hyperbolic manifolds

Abstract

We show that in every commensurability class of cusped arithmetic hyperbolic manifolds of simplest type of dimension $2n+2\geq 6$ there are manifolds $M$ such that the Stiefel-Whitney classes $w_{2j}(M)$ are non-vanishing for all $0 \leq 2j \leq n$. We also show that for the same commensurability classes there are manifolds (different from the previous ones) that do not admit a $\text{spin}^\mathbb{C}$ structure.