Tangent bundles of Hantzsche–Wendt manifolds

We formulate a condition for the existence of a SpinCSpinC-structure on an oriented flat manifold MnMn with H2(Mn,R)=0H2(Mn,R)=0. We prove that MnMn has a SpinCSpinC-structure if and only if there exists a homomorphism ϵ:π1(Mn)→SpinC(n)ϵ:π1(Mn)→SpinC(n) such that λ̄n∘ϵ=h, where h:π1(Mn)→SO(n)h:π1(Mn)→SO(n) is a holonomy homomorphism and λ̄n:SpinC(n)→SO(n) is a standard homomorphism defined. As an application we shall prove that all cyclic Hantzsche–Wendt manifolds do not have the SpinCSpinC-structure.