Online seminar, U. Haifa Topology & Geometry seminar on June 13, 2021
Topology & Geometry Seminar
Speaker: Alexander Berglund, Stockholm University
Topic: Characteristic classes of manifold bundles and graph homology
Place: This is an online seminar. Please email David Blanc “blanc at math dot haifa dot ac dot il” for the Zoom ID and password.
Date: Sunday, June 13, 2021
We introduce new families of rational characteristic classes of fiber
bundles with fiber a simply connected closed manifold. The new classes
are associated to homology classes in a certain graph complex. The graph
complex we consider is closely related to the Feynman transform of the
Lie operad, the homology of which is known to be expressible in terms of
the homology of automorphism groups of free groups.
The new classes may be viewed as a generalization of the
Miller-Morita-Mumford classes; these are recovered from degree zero
graph homology. When specialized to block bundles with fiber
#^g S^d x S^d (the g-fold connected sum of S^d x S^d), we recover
certain classes whose existence was implied by the calculation, due to
Ib Madsen and myself, of the stable rational cohomology of the block
diffeomorphism group (relative to a disk) of #^g S^d x S^d (2d>4). In
particular, this implies that the family of characteristic classes
associated to a non-trivial graph homology class is non-trivial
Our approach is based on rational homotopy theory and entails the
construction of new rational models for classifying spaces of fibrations.