Characteristic classes of spin surface bundles. Appliations of the Madsen-Weiss theory

Details Characteristic classes of spin surface bundles. Appliations of the Madsen-Weiss theory Characteristic classes of spin surface bundles. Appliations of the Madsen-Weiss theory

2006-11-20

arxiv.org/abs/math/0611612v1

Mathematics

Algebraic Topology

95 pages. The authors PhD thesis. To be printed as "Bonner Mathematische Schriften", volume 381

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Characteristic classes of spin surface bundles. Appliations of the Madsen-Weiss theory does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Characteristic classes of spin surface bundles. Appliations of the Madsen-Weiss theory, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Characteristic classes of spin surface bundles. Appliations of the Madsen-Weiss theory will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-120824