The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2, ℤ) as well as their analogues in the 3-strand braid group with local coefficients in symmetric powers of the natural symplectic representation. The resulting answers (1) correspond to certain modular forms in characteristic zero and (2) the cohomology of certain spaces in homotopy theory in characteristic p. The torsion is given in terms of the structure of a ‘p-divided power algebra’. The work is an extension of the work in Callegaro et al. [The cohomology of the braid group B3 and of SL2(ℤ) with coefficients in a geometric representation, Quart. J. Math. 64 (2013), 847–889] as well as extensions of a classical computation of Shimura to integral coefficients. The results here contrast the local coefficients such as that in Looijenga [Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel–Jacobi map, J. Algebraic Geom. 5 (1996), 135–150] and Tillmann [The representation of the mapping class group of a surface on its fundamental group in stable homology, Quart. J. Math. 61 (2010), 373–380].

COHOMOLOGY OF BRAIDS, PRINCIPAL CONGRUENCE SUBGROUPS AND GEOMETRIC REPRESENTATIONS

CALLEGARO, FILIPPO GIANLUCA;SALVETTI, MARIO
2014-01-01

Abstract

The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2, ℤ) as well as their analogues in the 3-strand braid group with local coefficients in symmetric powers of the natural symplectic representation. The resulting answers (1) correspond to certain modular forms in characteristic zero and (2) the cohomology of certain spaces in homotopy theory in characteristic p. The torsion is given in terms of the structure of a ‘p-divided power algebra’. The work is an extension of the work in Callegaro et al. [The cohomology of the braid group B3 and of SL2(ℤ) with coefficients in a geometric representation, Quart. J. Math. 64 (2013), 847–889] as well as extensions of a classical computation of Shimura to integral coefficients. The results here contrast the local coefficients such as that in Looijenga [Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel–Jacobi map, J. Algebraic Geom. 5 (1996), 135–150] and Tillmann [The representation of the mapping class group of a surface on its fundamental group in stable homology, Quart. J. Math. 61 (2010), 373–380].
2014
Callegaro, FILIPPO GIANLUCA; F. R., Cohen; Salvetti, Mario
File in questo prodotto:
File Dimensione Formato  
2014quarterly.pdf

accesso aperto

Descrizione: articolo
Tipologia: Documento in Post-print
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 309.48 kB
Formato Adobe PDF
309.48 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/363868
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact