Characteristic classes of transformation groups

Topology of Lie Groups and Characteristic Classes

Characteristic Classes and Representations of Discrete Subgroups of Lie Groups

A volume invariant is used to characterize those representations of a countable group into a connected semisimple Lie group G which are injective and whose image is a discrete cocompact subgroup of G. Let IT be a discrete cocompact subgroup of G and consider the analytic variety Hom(7r, G) consisting of homomorphisms 0: TT —> G. Denote by K a maximal compact subgroup of G and X = K\G the associ...

Quantum Characteristic Classes and Energy Flow on Loop Groups

In [13] we defined characteristic cohomology classes of the free loop space LHam(M, ω) and have shown that these classes give rise to a graded ring homomorphism Ψ : H∗(LHam(M, ω), Q)→ QH∗+2n(M). Here we explore connections of the map Ψ with Kähler geometry and energy flow on the loop spaces of compact Lie groups. Using this, we partially compute the composition Ψ : H∗(ΩG) → H∗(ΩHam(G/T )) → QH∗...

Mapping Class Groups, Characteristic Classes and Bernoulli Numbers

The mapping class group Γg of a closed, connected and oriented surface Sg of genus g is defined as the group of connected components of the group of orientation preserving diffeomorphisms of Sg. This group has been the object of many recent studies. Of particular interest are its finite subgroups; these are precisely the finite groups which occur as groups of symmetries of the surface Sg equipp...

Characteristic Classes

These are lecture notes for a series of five lectures I gave to other graduate students about characteristic classes through UT Austin’s summer minicourse program (see https://www.ma.utexas.edu/users/ richard.wong/Minicourses.html for more details). Beware of potential typos. In these notes I cover the basic theory of Stiefel-Whitney, Wu, Chern, Pontrjagin, and Euler classes, introducing some i...