Characteristic Classes of Spin Surface Bundles: Applications of the Madsen-Weiss Theory
Year: 2006
Type: dissertation
Abstract: In this work, we study topological properties of surface bundles, with an emphasis on surface bundles with a spin structure. We develop a criterion to decide whether a given manifold bundle has a spin structure and specialize it to surface bundles. We study examples of surface bundles, in particular sphere and torus bundles and surface bundles induced by actions of finite groups on Riemann surfaces. The examples are used to show that the obstruction cohomology class against the existence of a spin structure is nonzero. We develop a connection between the Atiyah-Singer index theorem for families of elliptic operators and the modern homotopy theory of moduli spaces of Riemann surfaces due to Tillmann, Madsen and Weiss. This theory and the index theorem is applied to prove that the tautological classes of spin surface bundles satisfy certain divisibility relations. The result is that the divisibility improves, compared with the non-spin-case, by a certain power of 2. The explicit computations for sphere bundles are used in the proof of the divisibility result. In the last chapter, we use actions of certain finite groups to construct explicit torsion elements in the homotopy groups of the mapping class and compute their order, which relies on methods from algebraic $K$-theory and on the Madsen-Weiss theorem.
Author: Johannes Ebert
Institution: University of Bonn
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1773326906