Finite complex reflection arrangements are K(π, 1)

• 成果类型: Article
• 署名作者: Bessis, David
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2015.181.3.1
• 发表日期: 2015
• 页码: 809-904
• 关键词: non-crossing partitions; regular elements; invariant-theory; artin groups; braid-groups; discriminants; HOMOLOGY; variety

摘要

Let V be a finite dimensional complex vector space and W subset of GL( V) be a finite complex reflection group. Let V-reg be the cornplemrnent in V of the reflecting hyperplanes. We prove that Vreg is a K(pi, 1) space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection groups. The complexified real case follows from a theorem of Deligne and, after contributions by Nakamura and Orlik-Solomon, only six exceptional cases remained open. In addition to solving these six cases, our approach is applicable to most previously known cases, including cornplexified real groups for which we obtain a new proof, based on new geometric objects. We also address a number of questions about pi(1)(W\V-reg), the braid group of W. This includes a description of periodic elements in terms of a braid analog of Springer's theory of regular elements.