Speaker: Rafał Lutowski (Gdańsk)
Title: Spin\(^c\) Structures on Real Bott Manifolds
Abstract:
The study of spin structures on manifolds has provided significant insights into their geometry and topology. One can consider the related, yet distinct, problem of the existence of spin\(^c\) structures. An orientable manifold admits a spin\(^c\) structure if its second Stiefel-Whitney class is the mod-2 reduction of an integral class, a less restrictive condition than that for spin structures. This makes spin\(^c\) structures a natural object of study for manifolds that are not spin.
In the talk we focus our attention on real Bott manifolds. They are constructed as iterated \(\mathbb{R}P^1\)-bundles, and are particularly amenable to a combinatorial approach. Their structure can be encoded by a strictly upper binary matrix, which also governs their cohomology. Building on this combinatorial framework, we establish a precise criterion for the existence of spin\(^c\) structures on any real Bott manifold. This result will be presented, and we will discuss how it generalizes and complements the known criteria for the existence of spin structures.
This is a joint work with Anna Gąsior.