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Proof of the K(π,1) conjecture for affine Artin groups

Paolini, Giovanni and Salvetti, Mario (2021) Proof of the K(π,1) conjecture for affine Artin groups. Inventiones Mathematicae, 224 (2). pp. 487-572. ISSN 0020-9910. doi:10.1007/s00222-020-01016-y.

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We prove the K(π,1) conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol’d, Pham, and Thom. Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on several new results and constructions. In particular: we show that all affine noncrossing partition posets are EL-shellable; we use these posets to construct finite classifying spaces for dual affine Artin groups; we introduce new CW models for the orbit configuration spaces associated with arbitrary Coxeter groups; we construct finite classifying spaces for the braided crystallographic groups introduced by McCammond and Sulway.

Item Type:Article
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URLURL TypeDescription ReadCube access Paper
Paolini, Giovanni0000-0002-3964-9101
Salvetti, Mario0000-0002-4820-5326
Additional Information:© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit Received 03 November 2019; Accepted 14 October 2020; Published 25 November 2020. We are grateful to Pierre Deligne for his remarks and suggestions on the first version of this paper. We are also grateful to Emanuele Delucchi and Alessandro Iraci for the useful discussions, and to the anonymous referee for the helpful comments. A preliminary version of Sects. 5, 6 and 7 is part of the first author’s Ph.D. thesis at Scuola Normale Superiore [56], written under the supervision of the second author. This work was also supported by the Swiss National Science Foundation Professorship Grant PP00P2_179110/1, by Ministero dell’Istruzione, dell’Università e della Ricerca, Prog. PRIN 2017YRA3LK_005, Moduli and Lie Theory and by the University of Pisa, Prog. PRA_2018_22, Geometria e topologia delle varietà. Open access funding provided by University of Fribourg.
Funding AgencyGrant Number
Swiss National Science Foundation (SNSF)PP00P2_179110/1
Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR)PRIN 2017YRA3LK_005
University of PisaPRA_2018_22
Université de FribourgUNSPECIFIED
Issue or Number:2
Classification Code:Mathematics Subject Classification: 20F36; 20F55; 55R35
Record Number:CaltechAUTHORS:20201214-124222536
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Official Citation:Paolini, G., Salvetti, M. Proof of the K(\pi , 1) conjecture for affine Artin groups. Invent. math. 224, 487–572 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:107071
Deposited By: Tony Diaz
Deposited On:14 Dec 2020 20:53
Last Modified:13 Apr 2021 22:07

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