Spin-structures on real Bott manifolds

Abstract

Let $$M_n\stackrel{\mathbb{R}{P}^1}{\to }{M}_{n-1}\stackrel{\mathbb{R}{P}^1}{\to }\dots \stackrel{\mathbb{R}{P}^1}{\to }{M}_1\stackrel{\mathbb{R}{P}^1}{\to }{M}_0=\{\bullet \}$$ be a sequence of real projective bundles such that $M_i\to M_{i-1}$, $i=1,2,\dots ,n$, 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 $\epsilon :\Gamma \to \mathrm{Spin}(n)$ such that ${\lambda }_n\epsilon =p$, 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. Gąsior, A. Szczepański, 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 $1\le i<j\le n$ 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.

After referee's remarks

Cited by in corpus (2)

• Spin structures of flat manifolds of diagonal type
• Some results on the topology of real Bott towers