76 Pages
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Higher complements of combinatorial sphere arrangements

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76 Pages
English

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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


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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 Osnabr¨uck,October8,2009 Nice, October 15, 2009
Higher complements of combinatorial sphere arrangements
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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.