Higher complements of combinatorial sphere arrangements

Higher complements of combinatorial sphere arrangements

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Higher complements of combinatorial sphere arrangements Higher complements of combinatorial sphere arrangements Clemens Berger University of Nice Combinatorial Structures in Algebra and Topology Osnabruck, October 8, 2009 Nice, October 15, 2009

coxeter arrangement

oriented matroids

acts simply

arrangement

topology osnabruck

??a h?

higher complements

euclidean space

Publié par

Nombre de lectures

6

Langue

English

Berger Arrangement Euclidean space

Higher complements of combinatorial sphere arrangements

Higher complements of combinatorial sphere arrangements

Clemens Berger

University of Nice

Combinatorial Structures in Algebra and Topology Osnabr¨uck,October8,2009 Nice, October 15, 2009

Higher complements of combinatorial sphere arrangements

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2

3

4

Hyperplane arrangements

Oriented matroids

Higher Salvetti complexes

The adjacency graph

Higher complements of combinatorial sphere arrangements Hyperplane arrangements

A (central)hyperplane arrangementAin euclidean spaceVis a ﬁnite family (Hα)α∈Aof hyperplanes ofVcontaining the origin. The arrangement isessentialif its centerTα∈AHαis trivial. ThecomplementM(A) =V\(Sα∈AHα) decomposes into path components, calledchambers(ortopes):CA=π0(M(A)). Denote bysαtheorthogonal symmetrywith respect toHα. If (Hα)α∈Ais stable undersβfor allβ∈ A, the arrangement is called aCoxeter arrangement write. WeA=AWwhereWis the subgroupW=<sα, α∈ A>ofOn(R is justiﬁed by). This

Proposition (Coxeter,Tits) There is a one-to-one correspondence between essential Coxeter arrangementsAWand ﬁnite Coxeter groupsW latter are. The classiﬁed by their Coxeter diagrams.

The Coxeter groupWacts simply transitively onCAW.

Higher complements of combinatorial sphere arrangements Hyperplane arrangements

A (central)hyperplane arrangementAin euclidean spaceVis a ﬁnite family (Hα)α∈Aof hyperplanes ofVcontaining the origin. The arrangement isessentialif its centerTα∈AHαis trivial. ThecomplementM(A) =V\(Sα∈AHα) decomposes into path components, calledchambers(ortopes):CA=π0(M(A)). Denote bysαtheorthogonal symmetrywith respect toHα. If (Hα)α∈Ais stable undersβfor allβ∈ A, the arrangement is called aCoxeter arrangement. We writeA=AWwhereWis the subgroupW=<sα, α∈ A>ofOn(R is justiﬁed by). This

Proposition (Coxeter,Tits) There is a one-to-one correspondence between essential Coxeter arrangementsAWand ﬁnite Coxeter groupsW latter are. The classiﬁed by their Coxeter diagrams.

The Coxeter groupWacts simply transitively onCAW.

Higher complements of combinatorial sphere arrangements Hyperplane arrangements

A (central)hyperplane arrangementAin euclidean spaceVis a ﬁnite family (Hα)α∈Aof hyperplanes ofVcontaining the origin. The arrangement isessentialif its centerTα∈AHαis trivial. ThecomplementM(A) =V\(Sα∈AHα) decomposes into path components, calledchambers(ortopes):CA=π0(M(A)). Denote bysαtheorthogonal symmetrywith respect toHα. If (Hα)α∈Ais stable undersβfor allβ∈ A, the arrangement is called aCoxeter arrangement write. WeA=AWwhereWis the subgroupW=<sα, α∈ A>ofOn(R is justiﬁed by). This

Proposition (Coxeter,Tits) There is a one-to-one correspondence between essential Coxeter arrangementsAWand ﬁnite Coxeter groupsW latter are. The classiﬁed by their Coxeter diagrams.

The Coxeter groupWacts simply transitively onCAW.

Higher complements of combinatorial sphere arrangements Hyperplane arrangements

A (central)hyperplane arrangementAin euclidean spaceVis a ﬁnite family (Hα)α∈Aof hyperplanes ofVcontaining the origin. The arrangement isessentialif its centerTα∈AHαis trivial. ThecomplementM(A) =V\(Sα∈AHα) decomposes into path components, calledchambers(ortopes):CA=π0(M(A)). Denote bysαtheorthogonal symmetrywith respect toHα. If (Hα)α∈Ais stable undersβfor allβ∈ A, the arrangement is called aCoxeter arrangement. We writeA=AWwhereWis the subgroupW=<sα, α∈ A>ofOn(R is justiﬁed by). This

Proposition (Coxeter,Tits) There is a one-to-one correspondence between essential Coxeter arrangementsAWand ﬁnite Coxeter groupsW. The latter are classiﬁed by their Coxeter diagrams.

The Coxeter groupWacts simply transitively onCAW.

Higher complements of combinatorial sphere arrangements Hyperplane arrangements

A (central)hyperplane arrangementAin euclidean spaceVis a ﬁnite family (Hα)α∈Aof hyperplanes ofVcontaining the origin. The arrangement isessentialif its centerTα∈AHαis trivial. ThecomplementM(A) =V\(Sα∈AHα) decomposes into path components, calledchambers(ortopes):CA=π0(M(A)). Denote bysαtheorthogonal symmetrywith respect toHα. If (Hα)α∈Ais stable undersβfor allβ∈ A, the arrangement is called aCoxeter arrangement. We writeA=AWwhereWis the subgroupW=<sα, α∈ A>ofOn(R is justiﬁed by). This

Proposition (Coxeter,Tits) There is a one-to-one correspondence between essential Coxeter arrangementsAWand ﬁnite Coxeter groupsW. The latter are classiﬁed by their Coxeter diagrams.

The Coxeter groupWacts simply transitively onCAW.

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