Spin-structures on real bott manifolds

TLDR: Sior et al. as discussed by the authors formulated necessary and sufficient conditions of the existence of Spin-structure on real Bott manifolds. But they did not consider the case where k is even.

Abstract: Let $$M_{n}\\stackrel{\\mathbb R P^1}\\to M_{n-1}\\stackrel{\\mathbb R P^1}\\to\\ldots\\stackrel{\\mathbb R P^1}\\to M_{1}\\stackrel{\\mathbb R P^1}\\to M_0 = \\{ \\bullet\\} $$ 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. \nThere 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 $\\epsilon\\colon\\Gamma\\to\\operatorname{Spin}(n)$ such that $\\lambda_n\\epsilon=p$, where $\\lambda_n:\\operatorname{Spin}(n)\\to\\operatorname{SO}(n)$ 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. \nOur 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 \nThe real Bott manifold $M(A)$ has a Spin-structure if and only for all $1\\leq i