Tangent bundles of Hantzsche-Wendt manifolds

TL;DR

This paper establishes a criterion for the existence of a $Spin^C$-structure on oriented manifolds with trivial second cohomology and applies it to show that all cyclic Hantzsche-Wendt manifolds lack such structures.

Contribution

It introduces a new condition for $Spin^C$-structure existence and demonstrates its application to a class of Hantzsche-Wendt manifolds.

Findings

All cyclic Hantzsche-Wendt manifolds do not admit $Spin^C$-structures.

A specific cohomological condition determines $Spin^C$-structure existence.

The criterion applies to oriented manifolds with vanishing second cohomology.

Abstract

We formulate a condition for an existence of a $Spin^C$ - structure on an oriented at manifold $M^n$ with $H^2(Mn;R) = 0$. As an application we shall prove that all cyclic Hantzsche - Wendt manifolds have not the $Spin^C$-structure.