The main aim of this article is to give a necessary and sufficient condition for a real Bott manifold to admit a spin structure and further give a combinatorial characterization for the spin structure in terms of the associated acyclic digraph.

On the topology of real Bott manifolds

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. 

https://journals.tubitak.gov.tr/cgi/viewcontent.cgi?article=3420&context=math

Spin structures on generalized real Bott manifolds

On the Topology of Real Bott Manifolds

Let M be a real Bott manifold with K\"{a}hler structure. Using Ishida characterization we give necessary and sufficient condition for the existence of the Spin-structure on M. In proof we use the technic developed in Popko, Szczepa\'{n}ski "Cohomological rigity of oriented Hantzsche-Wendt manifolds" (Adv. Math. 302 (2016), 1044 - 1068) and characteristic classes.

Spin-structures on real Bott manifolds with K\"ahler structures

Generalized Bott manifolds (over $\mathbb C$ and $\mathbb R$) have been defined by Choi, Masuda and Suh. In this article we extend the results of arXiv:1609.05630 on the topology of real Bott manifolds to generalized real Bott manifolds. We give a presentation of the fundamental group, prove that it is solvable and give a characterization for it to be abelian. We further prove that these manifolds are aspherical only in the case of real Bott manifolds and compute the higher homotopy groups. Furthermore, using the presentation of the cohomology ring with $\mathbb Z_2$-coefficients, we derive a combinatorial characterization for orientablity and spin structure.

Results on the topology of generalized real Bott manifolds

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

Anna Gasior

Papers

Spin-structures on real bott manifolds