%0 Journal Article %T Transgressions of the Euler class and Eisenstein cohomology of GLN(Z) %+ Département de Mathématiques et Applications - ENS Paris (DMA) %+ Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)) %+ Department of Mathematics (UCL London) %A Bergeron, Nicolas %A Charollois, Pierre %A Garcia, Luis %< avec comité de lecture %@ 0289-2316 %J Japanese Journal of Mathematics %I Springer Verlag %V 15 %N 2 %P 311-379 %8 2020-09 %D 2020 %R 10.1007/s11537-019-1822-6 %Z Mathematics [math]/Number Theory [math.NT] %Z Mathematics [math]/Geometric Topology [math.GT]Journal articles %X 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 , 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. %G English %2 https://ens.hal.science/hal-02886362/document %2 https://ens.hal.science/hal-02886362/file/Takagi-final-JJM1822.pdf %L hal-02886362 %U https://ens.hal.science/hal-02886362 %~ ENS-PARIS %~ CNRS %~ INSMI %~ IMJ %~ PSL %~ SORBONNE-UNIVERSITE %~ SORBONNE-UNIV %~ SU-SCIENCES %~ MATH_ENS_PARIS %~ UNIV-PARIS %~ UNIVERSITE-PARIS %~ UP-SCIENCES %~ ENS-PSL %~ SU-TI %~ ALLIANCE-SU