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Topology of Moduli Spaces and Strings

Final Report Summary - TMSS (Topology of Moduli Spaces and Strings)

The homological and homotopical structure of a number of moduli spaces has been studied, including the moduli space E(M) of smooth submanifolds of infinite dimensional euclidian space and the moduli space KR(A,a) of pairs (P,b) of a projective module equipped with a bilinear form over a ring A with an anti-involution a:A-->A.

Main results:
1. For M a (d-1) connected 2d dimensional manifold, S. Galatius and O. Randal-Williams determined the cohomology of E(M) in a "stable" range of dimensions. A. Berglund and the PI calculated the corresponding moduli space of blocked embedded manifolds. A comparison between these two results is a new challenge of considerable interest.

2. Lars Hesselholt and the PI has defined and examined the moduli space KR(A,a). This is a Real version of algebraic K-theory with a conjugation operator. Its fixed set is the Grothendieck-Witt space examined by M. Schlichting. We prove that it is a group completion of the stable automorphism group of the pairs (P,b) of a projective module equipped with a bilinear form over (A,a).

3. The PI proved a Riemann-Roch theorem for surface bundles, solving a conjecture of T. Akita.

The project has supported a total of six Post docs, each hired for one or two years, including Randal-Williams and Berglund. Moreover the project has supported short time visitors, including Galatius and Hesselholt who have visited one or two months per year.

All Post Docs that have been supported by the project are now hired in university jobs elsewhere. The PI has been advisor for four PhD students of which one has graduated and the rest will graduate in the following years.