Annals of Mathematics· 2014Q1
Finite complex reflection arrangements are K(pi,1)
- 132citations
- Q1SCImago
- 2014year
Short summary
The complement of reflecting hyperplanes in a complex vector space, V reg, is proven to be a K(π,1) space, resolving a long-standing conjecture.
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Key points
- The complement of reflecting hyperplanes in a complex vector space (V reg) is demonstrated to be a K(π,1) space.
- This result resolves the final six open cases of the Brieskorn conjecture for complex reflection groups.
- A new geometric approach provides a novel proof for complexified real reflection groups.
- The study characterizes periodic elements in the braid group of W using a braid analog of Springer's theory.
AI-generated from the title and abstract; the full text is not read.
Abstract
Let V be a finite dimensional complex vector space and W GL(V ) be a finite complex reflection group. Let V reg be the complement in V of the reflecting hyperplanes. We prove that V reg is a K(, 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 complexified real groups for which we obtain a new proof, based on new geometric objects. We also address a number of questions about 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.
The authors' abstract, as published at the source. Annals of Mathematics, 2014 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics