Cohomology of braids, principal congruence subgroups and geometric\n representations

2012/07/24 by Filippo Callegaro, Fred Cohen, Callegaro, Filippo +3
Mathematics · #20J06 #55N25 #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1207.5652

openalex publication_date 2012/07/24 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

The main purpose of this article is to give the integral cohomology of\nclassical principal congruence subgroups in SL(2,Z) as well as their analogues\nin the third braid group with local coefficients in symmetric powers of the\nnatural symplectic representation. The resulting answers (1) correspond to\ncertain modular forms in characteristic zero, and (2) the cohomology of certain\nspaces in homotopy theory in characteristic p. The torsion is given in terms of\nthe structure of a "p-divided power algebra".\n The work is an extension of the work in arXiv:1204.5390v1 as well as\nextensions of a classical computation of Shimura to integral coefficients. The\nresults here contrast the local coefficients such as that in [Looijenga, J.\nAlg. Geom., 5, '96] and [Tillmann, Q. J. Math., 61, '10].\n

Citations

• Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map

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