Spin-structures and 2-fold coverings (original) (raw)

Spin Structures on Compact Homogeneous Pseudo-Riemannian Manifolds

Transformation Groups, 2018

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds F = G/H, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M = G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin.

Spin Structures on Generalized Real Bott Manifolds

2021

Bk −−−→ Bk−1 −−−→ · · · −−−→ B1 −−−→ {pt} (1) where Bi is the projectivization of the Whitney sum of ni + 1 real line bundles over Bi−1. This notion is introduced by Choi, Masuda and Suh [4] as a generalization of the notion of a Bott tower given in [9]. The manifold Bk is called a real Bott manifold when ni = 1 for each i and a generalized real Bott manifold, otherwise. The manifold Bk can be realized as a small cover over ∏k i=1∆ ni where ∆i is the ni-simplex [10, Corollary 4.6]. It is also known that every small cover over a product of simplices is a generalized real Bott manifold [4, Remark 6.5]. When P is a simple convex polytope of dimension n with the facet set F(P ) = {F1, · · · , Fm}, for every small cover M over P , there is an associated (n × (m − n)) matrix A = [aij ] with entries in Z2 which can be used to reconstruct M (See Section 2). Moreover, the mod 2 cohomology ring structure of M depends only on the face poset of P and the matrix A. More precisely, let Z2[P ] be ...

Classification of Spin Structures on Four-Dimensional Almost-Flat Manifolds

Mathematika, 2018

Almost-flat manifolds were defined by Gromov as a natural generalization of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental group. Utilizing this, we classify the spin structures on all four-dimensional almost-flat manifolds that are not flat. Out of 127 orientable families, we show that there are exactly 15 that are non-spin, the rest are, in fact, parallelizable.

Spin structures of flat manifolds of diagonal type

Homology, Homotopy and Applications

For each integer d at least two, we construct non-spin closed oriented flat manifolds with holonomy group Z d 2 and with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have trivial Stiefel-Whitney classes.

The Moduli Space of Principal Spin Bundles´alvaro

2016

We study the geometry of the moduli space of principal Spin bundles through the study of the subvarieties of fixed points of automorphisms of finite order coming from outer automorphisms of the structure group. We describe the nonstable and the singular locus of the moduli space, prove that nonstable bundles are singular points, and identify the fixed points, in the case of M (Spin(8, C)), in the stable or strictly polystable locus of the moduli space.

Spin Structures on Flat Manifolds

Monatshefte für Mathematik, 2006

The aim of this paper is to present some results about spin structures on flat manifolds. We prove that any finite group can be the holonomy group of a flat spin manifold. Moreover we shall give some methods of constructing spin structures related to the holonomy representation.

Algebra and Structures of Spinors Fiber Bundles

Omdurman Islamic University Journal, 2015

The aim of this paper is to investigate the mathematics of spinor bundles and their classification. We devote the methods of principal fiber bundles allows through a coherent treatment of Pseudo-Riemannian manifolds and spinor structures with Clifford algebras which couple to Dirac operator to study important applications in cohomology theory.