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Cohomology of Artin groups of type Ãn, Bn and applications

Filippo Callegaro, Davide Moroni and Mario Salvetti

Geometry & Topology Monographs 13 (2008) 85–104

DOI: 10.2140/gtm.2008.13.85

arXiv: 0904.0457

Abstract

We consider two natural embeddings between Artin groups: the group GÃn-1 of type Ãn-1 embeds into the group GBn of type Bn; GBn in turn embeds into the classical braid group Brn+1:=GAn of type An. The cohomologies of these groups are related, by standard results, in a precise way. By using techniques developed in previous papers, we give precise formulas (sketching the proofs) for the cohomology of GBn with coefficients over the module Q[q±1,t±1], where the action is (-q)–multiplication for the standard generators associated to the first n-1 nodes of the Dynkin diagram, while is (-t)–multiplication for the generator associated to the last node.

As a corollary we obtain the rational cohomology for GÃn as well as the cohomology of Brn+1 with coefficients in the (n+1)–dimensional representation obtained by Tong, Yang and Ma.

We stress the topological significance, recalling some constructions of explicit finite CW–complexes for orbit spaces of Artin groups. In case of groups of infinite type, we indicate the (few) variations to be done with respect to the finite type case. For affine groups, some of these orbit spaces are known to be K(π,1) spaces (in particular, for type Ãn).

We point out that the above cohomology of GBn gives (as a module over the monodromy operator) the rational cohomology of the fibre (analog to a Milnor fibre) of the natural fibration of K(GBn,1) onto the 2–torus.

Keywords

affine Artin groups, twisted cohomology, group representations

Mathematical Subject Classification

Primary: 20J06

Secondary: 20F36

References
Publication

Received: 30 May 2006
Revised: 17 January 2007
Accepted: 24 January 2007
Published: 22 February 2008

Authors
Filippo Callegaro
Scuola Normale Superiore
Piazza dei Cavalieri, 7
56126 Pisa
Italy
Davide Moroni
Dipartimento di Matematica “G.Castelnuovo”
Università di Roma “La Sapienza”
Piazza Aldo Moro, 2
00185 Roma
Italy
and
Istituto di Scienza e Tecnologie dell'Informazione ISTI-CNR
Via G. Moruzzi 1
56124 Pisa
Italy
Mario Salvetti
Dipartimento di Matematica “L.Tonelli”
Università di Pisa
Largo B. Pontecorvo, 5
56127 Pisa
Italy