Spin Structures on Generalized Real Bott Manifolds

TL;DR

This paper characterizes when generalized real Bott manifolds admit Spin structures using matrix and graph conditions, providing new criteria and examples of manifolds without Spin structures.

Contribution

It offers a necessary and sufficient condition for Spin structures on generalized real Bott manifolds based on matrix columns and graph interpretations.

Findings

Identifies criteria for Spin structures in terms of matrix columns

Provides a graph-theoretic interpretation of the conditions

Constructs examples of manifolds lacking Spin structures

Abstract

In this paper, we give a necessary and sufficient condition for a generalized real Bott manifold to have a Spin structure in terms of column vectors of the associated matrix. We also give an interpretation of this result to the associated acyclic $\omega$-weighted digraphs. Using this, we obtain a family of real Bott manifolds that does not admit Spin Structure.