Transgressions of the Euler class and Eisenstein cohomology of GL N (Z)

These notes were written to be distributed to the audience of the first author’s Takagi Lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh.In this work-in-progress we give a new construction of some Eisenstein classes for GL_N (Z) that were first considered by Nori [41] and Sczech [44]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of SL_N (Z) vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair (GL_N, GL₁). This suggests looking to reductive dual pairs (GL_N, GL_k) with k ≥ 1 for possible generalizations of the Eisenstein cocycle. This leads to fascinating lifts that relate the geometry/topology world of real arithmetic locally symmetric spaces to the arithmetic world of modular forms.In these notes we do not deal with the most general cases and put a lot of emphasis on various examples that are often classical.