Tangent bundles of Hantzsche-Wendt manifolds

Abstract

We formulate a condition for an existence of a $Spi{n}^C$ - 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 $Spi{n}^C$-structure.