https://ens.hal.science/hal-02886362Bergeron, NicolasNicolasBergeronDMA - Département de Mathématiques et Applications - ENS Paris - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris sciences et lettres - CNRS - Centre National de la Recherche ScientifiqueCharollois, PierrePierreCharolloisIMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris CitéGarcia, LuisLuisGarciaUCL London - Department of Mathematics - UCL - University College of London [London]Transgressions of the Euler class and Eisenstein cohomology of GLN(Z)HAL CCSD2020[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT][MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT]Laffont, Marine2020-07-01 14:38:062023-03-24 14:53:182020-07-02 08:38:09enJournal articleshttps://ens.hal.science/hal-02886362/document10.1007/s11537-019-1822-6application/pdf1These 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 , GL 1). 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 don't deal with the most general cases and put a lot of emphasis on various examples that are often classical.