Spin-structures on real Bott manifolds

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. 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 $\epsilon\colon\Gamma\to\operatorname{Spin}(n)$ 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. 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 $M(A_{ij})$ have a Spin-structure, where $A_{ij}$ are $n\times n$-integer matrices with $i-$th and $j-$th nonzero rows.