Higher complements of combinatorial sphere arrangements

Clemens Berger

Description

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

Sujets

• Berger
• Arrangement
• Euclidean space

Informations

• Publié par: pefav
• Nombre de lectures: 6
• Langue: English

Extrait

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 (H α ) α∈A of hyperplanes ofVcontaining the origin. The arrangement isessentialif its center T α∈A H α is trivial. ThecomplementM(A) =V\( S α∈A H α ) decomposes into path components, calledchambers(ortopes):C A =π 0 (M(A)). Denote bys α theorthogonal symmetrywith respect toH α . If (H α ) α∈A is stable unders β for allβ∈ A, the arrangement is called aCoxeter arrangement write. WeA=A W whereWis the subgroupW=<s α , α∈ A>ofO n (R is justified by). This

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

The Coxeter groupWacts simply transitively onC A W .

Higher complements of combinatorial sphere arrangements Hyperplane arrangements

A (central)hyperplane arrangementAin euclidean spaceVis a finite family (H α ) α∈A of hyperplanes ofVcontaining the origin. The arrangement isessentialif its center T α∈A H α is trivial. ThecomplementM(A) =V\( S α∈A H α ) decomposes into path components, calledchambers(ortopes):C A =π 0 (M(A)). Denote bys α theorthogonal symmetrywith respect toH α . If (H α ) α∈A is stable unders β for allβ∈ A, the arrangement is called aCoxeter arrangement. We writeA=A W whereWis the subgroupW=<s α , α∈ A>ofO n (R is justified by). This

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

The Coxeter groupWacts simply transitively onC A W .

Higher complements of combinatorial sphere arrangements Hyperplane arrangements

A (central)hyperplane arrangementAin euclidean spaceVis a finite family (H α ) α∈A of hyperplanes ofVcontaining the origin. The arrangement isessentialif its center T α∈A H α is trivial. ThecomplementM(A) =V\( S α∈A H α ) decomposes into path components, calledchambers(ortopes):C A =π 0 (M(A)). Denote bys α theorthogonal symmetrywith respect toH α . If (H α ) α∈A is stable unders β for allβ∈ A, the arrangement is called aCoxeter arrangement write. WeA=A W whereWis the subgroupW=<s α , α∈ A>ofO n (R is justified by). This

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

The Coxeter groupWacts simply transitively onC A W .

Higher complements of combinatorial sphere arrangements Hyperplane arrangements

A (central)hyperplane arrangementAin euclidean spaceVis a finite family (H α ) α∈A of hyperplanes ofVcontaining the origin. The arrangement isessentialif its center T α∈A H α is trivial. ThecomplementM(A) =V\( S α∈A H α ) decomposes into path components, calledchambers(ortopes):C A =π 0 (M(A)). Denote bys α theorthogonal symmetrywith respect toH α . If (H α ) α∈A is stable unders β for allβ∈ A, the arrangement is called aCoxeter arrangement. We writeA=A W whereWis the subgroupW=<s α , α∈ A>ofO n (R is justified by). This

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

The Coxeter groupWacts simply transitively onC A W .

Higher complements of combinatorial sphere arrangements Hyperplane arrangements

A (central)hyperplane arrangementAin euclidean spaceVis a finite family (H α ) α∈A of hyperplanes ofVcontaining the origin. The arrangement isessentialif its center T α∈A H α is trivial. ThecomplementM(A) =V\( S α∈A H α ) decomposes into path components, calledchambers(ortopes):C A =π 0 (M(A)). Denote bys α theorthogonal symmetrywith respect toH α . If (H α ) α∈A is stable unders β for allβ∈ A, the arrangement is called aCoxeter arrangement. We writeA=A W whereWis the subgroupW=<s α , α∈ A>ofO n (R is justified by). This

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

The Coxeter groupWacts simply transitively onC A W .