University of Chicago
Department of Mathematics

FIRST QUARTERLY AllChicago TOPOLOGY SEMINAR (joint between UIC, NU, and UC)
Abstract: One may construct an S^1-equivariant cohomology theory EC associated to a complex curve C of genus g with additional structure, so that the value of the cohomology theory on spheres of representations is equal to the sheaf cohomology of suitable line bundles on the curve. Having got the theory, it is natural to attempt to use theta functions to give Thom isomorphisms for suitable bundles. In the fairly familiar case when C is an elliptic curve (ie g=1), an extension of Ando's work shows that this works precisely if the Borel Chern classes c_1=c_2=0, and hence one obtains a rational S^1-equivariant sigma orientation MString--->EC, giving the Ando-Hopkins-Strickland sigma orientation non-equivariantly.

Fix a smooth manifold M and consider the space of pairs (S,f) where S is a Riemann surface and f is a smooth map S -> M. This space modulo an equivalence relation is the moduli space of smooth surfaces in M, MTop(M). MTop(M) is closely related to the moduli space of holomorphic curves studied in algebraic geometry.
A rational combinatorial model for MTop(M) is obtained from Igusa's Fat category. This category whose objects are fat graphs is a (rational) model for the moduli space of Riemann surfaces. Using Anderson and Serre spectral sequence, this combinatorial model for MTop(M) gives a combinatorial chain complex computing the rational homology of MTop(M) which generalizes the Kontsevich graph complex.

There has been much recent interest in constructing geometric objects, so-called `p-gerbes' or `p-line bundles', which realise classes in H^{p+2}(M;Z). For the case p=2 Brylinski and McLaughlin studied a certain class of 2-gerbes and developed differential geometric notions such as connections and curvature for these objects. 2-gerbes are category theoretic objects developed by Breen in his study of degree 3 non-abelian cohomology. In this talk we will introduce another class of geometric objects, `bundle 2-gerbes', which also realise classes in H^4(M;Z). We shall also describe a theory of connections and curvature for these objects and relate bundle 2-gerbes to the 2-gerbes of Brylinski and McLaughlin.

Cancelled