Spin-structures on real Bott manifolds

Let $$M_n{M}_{n-1}\dots M_1{M}_0=\{\bullet \}$$ be a sequence of real projective bundles such that $M_i\to M_{i-1}$,$i=1,2,\ldots,n$, is a projective bundle of a Whitney sum of a real line bundle $L_{i-1}$ and the trivial line bundle over $M_{i-1}$. The above sequence is called the real Bott tower and the top manifold $M_n$ is called the real Bott manifold. There are a few ways to decide whether there exists a Spin-structure on an oriented flat manifold $M^n$. An oriented flat manifold $M^n$ has a Spin-structure if and only if there exists a homomorphism $ϵ\text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\to \mathrm{Spin}(n)$ such that ${\lambda }_nϵ=p$, where ${\lambda }_n:\mathrm{Spin}(n)\to \mathrm{SO}(n)$ is the covering map. There is an equivalent condition for existence of Spin-structure. This is well known that the closed oriented differential manifold $M$ has a Spin-structure if and only if the second Stiefel-Whitney class vanishes. Our paper is a sequel of A. G\k{a}sior, A. Szczepa'nski, Flat manifolds with holonomy group $Z_2^k$ of diagonal type, Osaka J. Math. 51 (2014), 1015 - 1025. There are given non-complete conditions of the existence of Spin-structures on real Bott manifolds. In this paper, if k is even, we formulate necessary and sufficient conditions of the existence of Spin-structure on real Bott manifolds. Here is our main result The real Bott manifold $M(A)$ has a Spin-structure if and only for all $1\leq i<j\leq n$manifolds$M(A_{ij})$haveaSpin-structure,where$A_{ij}$are$n\times n$-integermatriceswith$i-$thand$j-$th nonzero rows.