Spin-Structures on Real Bott Manifolds

Abstract

Abstract. Real Bott manifolds is a class of flat manifolds with holonomy group Z k 2 of diagonal type. In this paper we formulate necessary and sufficient conditions of the existence of a Spin-structure on real Bott manifolds. It extends results of [9].

Spin structures of flat manifolds of diagonal type

“…Real Bott manifolds are a special type of flat manifolds with diagonal holonomy. By a result of A. Gąsior (see [7,Theorem 1.2]), it follows that a real Bott manifold with holonomy group of even Z 2rank has a spin structure if and only if all its finite covers with holonomy group Z 2 2 have a spin structure. Our examples show that the general case of diagonal flat manifolds is much more complicated.…”

On the Topology of Real Bott Manifolds

“…More precisely, in Proposition 4.1, we give a presentation with generators and relations of H * (B k ; Z 2 ). Further, we give a combinatorial criterion for B k to be orientable in Theorem 4.5 and for B k to be spin in the criterion for spin structure for real Bott manifolds given by Gaşior in [9] and the authors of the present paper in [8].…”