The Deligne Complex for the Four-strand Braid Group
نویسندگان
چکیده
This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on Cn. A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1).
منابع مشابه
Semi-algebraic Geometry of Braid Groups
The braid group of n-strings is the group of homotopy types of movements of n distinct points in the 2-plane R. It was introduced by E. Artin [1] in 1926 in order to study knots in R. He gave a presentation of the braid group by generators and relations, which are, nowadays, called the Artin braid relations. Since then, not only in the study of knots, the braid groups appear in several contexts...
متن کاملIrreducibility of the tensor product of Albeverio's representations of the Braid groups $B_3$ and $B_4$
We consider Albeverio's linear representations of the braid groups $B_3$ and $B_4$. We specialize the indeterminates used in defining these representations to non zero complex numbers. We then consider the tensor products of the representations of $B_3$ and the tensor products of those of $B_4$. We then determine necessary and sufficient conditions that guarantee the irreducibility of th...
متن کاملComplex Reflection Arrangements Are K ( Π , 1 )
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(π, 1) space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection group. The complexified real case follows from a theorem of Deligne and, after ...
متن کاملPresentations for the Cohomology Rings of Tree Braid Groups
If Γ is a finite graph and n is a natural number, then UCnΓ, the unlabelled configuration space of n points on Γ, is the space of all n-element subsets of Γ. The fundamental group of UCnΓ is the n-strand braid group of Γ, denoted BnΓ. We build on earlier work to compute presentations of the integral cohomology rings H∗(UCnT ;Z), where T is any tree and n is arbitrary. The results suggest that H...
متن کاملFinite Complex Reflection Arrangements Are K ( Π , 1 )
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(π, 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...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004