Centrally Symmetric Manifolds with Few Vertices Steven Klee joint - PowerPoint PPT Presentation

Background A New Construction Centrally Symmetric Manifolds with Few Vertices Steven Klee joint with Isabella Novik UC Davis AMS Spring Sectional Meeting University of Southern Georgia March 12, 2011 Steven Klee CS Manifolds with Few Vertices

Background A New Construction Background 1 Definitions Products of Spheres Known Results A New Construction 2 Our Result The Construction Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results Centrally Symmetric Complexes Definition A pure ( d − 1)-dimensional simplicial complex ∆ is centrally symmetric if there is an involution ϕ : V (∆) → V (∆) such that for all faces F ∈ ∆, ϕ ( F ) ∈ ∆, and ϕ ( F ) � = F . Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results CS Neighborly Complexes d := conv {± e 1 , ± e 2 , . . . , ± e d } ⊆ R d is centrally Example: C ∗ symmetric under the involution ϕ ( e i ) = − e i . x 3 y 2 x 1 x 2 y 1 y 3 Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results CS Neighborly Complexes Definition A centrally symmetric simplicial complex ∆ on 2 d vertices is (cs) k-neighborly if it has the k -skeleton of C ∗ d . Example: A cs 1-neighborly triangulation of S 1 × S 1 . x 1 y 3 y 1 x 3 x 1 x 2 y 4 x 2 y 2 x 4 y 4 x 1 y 3 y 1 x 3 x 1 Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results CS Products of Spheres Proposition A cs triangulation of S i × S d − i − 2 requires at least 2 d vertices. Question Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results CS Products of Spheres Proposition A cs triangulation of S i × S d − i − 2 requires at least 2 d vertices. Question Do there exist cs triangulations of S i × S d − i − 2 with exactly 1 2 d vertices? Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results CS Products of Spheres Proposition A cs triangulation of S i × S d − i − 2 requires at least 2 d vertices. Question Do there exist cs triangulations of S i × S d − i − 2 with exactly 1 2 d vertices? (Sparla) Are there i -neighborly triangulations of S i × S i with 2 exactly 4 i + 4 vertices? Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results CS Products of Spheres Proposition A cs triangulation of S i × S d − i − 2 requires at least 2 d vertices. Question Do there exist cs triangulations of S i × S d − i − 2 with exactly 1 2 d vertices? (Sparla) Are there i -neighborly triangulations of S i × S i with 2 exactly 4 i + 4 vertices? 2 ⌉ × S ⌊ d 2 ⌋ -neighborly triangulations of S ⌈ d (Lutz) Are there ⌊ d 2 ⌋ 3 on 2 d + 4 vertices that admit a dihedral group action? Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results Known Results Question 1: Do there exist cs triangulations of S i × S d − i − 2 with exactly 2 d vertices? Theorem (K¨ uhnel-Lassmann) There exists a cs triangulation of S 1 × S d − 3 on 2 d vertices for all d ≥ 3 . (Lutz) Such triangulations exist for all d ≤ 10 (computer check). Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results Known Results Question 2: (Sparla) Are there i -neighborly triangulations of S i × S i with exactly 4 i + 4 vertices? Examples Triangulation of S 1 × S 1 on 8 vertices. Triangulation of S 2 × S 2 on 12 vertices due to Sparla (’97). Effenberger (’09) proposed a construction for all i ; verified by computer for all i ≤ 13. Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results Known Results Question 3: Are there ⌊ d 2 ⌋ -neighborly triangulations of 2 ⌉ × S ⌊ d 2 ⌋ on 2 d + 4 vertices that admit a dihedral group action? S ⌈ d Theorem (Lutz) When d ≤ 8, there are cs triangulations of S i × S d − i − 2 that admit a vertex-transitive action by a dihedral group of order 4 d Steven Klee CS Manifolds with Few Vertices

Definitions Background Products of Spheres A New Construction Known Results Known Results Question 3: Are there ⌊ d 2 ⌋ -neighborly triangulations of 2 ⌉ × S ⌊ d 2 ⌋ on 2 d + 4 vertices that admit a dihedral group action? S ⌈ d Theorem (Lutz) When d ≤ 8, there are cs triangulations of S i × S d − i − 2 that admit a vertex-transitive action by a dihedral group of order 4 d EXCEPT in the cases of S 2 × S 4 and S 2 × S 6 . Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction Our Main Result Theorem (K.-Novik, ’11) For all 0 ≤ i ≤ d − 1, there exists combinatorial manifold B ( i , d ) satisfying the following properties: B ( i , d ) is centrally symmetric on 2 d vertices; 1 Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction Our Main Result Theorem (K.-Novik, ’11) For all 0 ≤ i ≤ d − 1, there exists combinatorial manifold B ( i , d ) satisfying the following properties: B ( i , d ) is centrally symmetric on 2 d vertices; 1 B ( i , d ) is cs i -neighborly; 2 Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction Our Main Result Theorem (K.-Novik, ’11) For all 0 ≤ i ≤ d − 1, there exists combinatorial manifold B ( i , d ) satisfying the following properties: B ( i , d ) is centrally symmetric on 2 d vertices; 1 B ( i , d ) is cs i -neighborly; 2 H ∗ ( B ( i , d ); Z ) ≈ � � H ∗ ( S i ; Z ); 3 Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction Our Main Result Theorem (K.-Novik, ’11) For all 0 ≤ i ≤ d − 1, there exists combinatorial manifold B ( i , d ) satisfying the following properties: B ( i , d ) is centrally symmetric on 2 d vertices; 1 B ( i , d ) is cs i -neighborly; 2 H ∗ ( B ( i , d ); Z ) ≈ � � H ∗ ( S i ; Z ); 3 ∂ B ( i , d ) triangulates S i × S d − i − 2 ; 4 Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction Our Main Result Theorem (K.-Novik, ’11) For all 0 ≤ i ≤ d − 1, there exists combinatorial manifold B ( i , d ) satisfying the following properties: B ( i , d ) is centrally symmetric on 2 d vertices; 1 B ( i , d ) is cs i -neighborly; 2 H ∗ ( B ( i , d ); Z ) ≈ � � H ∗ ( S i ; Z ); 3 ∂ B ( i , d ) triangulates S i × S d − i − 2 ; 4 ∂ B ( i , d ) is cs i -neighborly when i ≤ d − i − 2; 5 Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction Our Main Result Theorem (K.-Novik, ’11) For all 0 ≤ i ≤ d − 1, there exists combinatorial manifold B ( i , d ) satisfying the following properties: B ( i , d ) is centrally symmetric on 2 d vertices; 1 B ( i , d ) is cs i -neighborly; 2 H ∗ ( B ( i , d ); Z ) ≈ � � H ∗ ( S i ; Z ); 3 ∂ B ( i , d ) triangulates S i × S d − i − 2 ; 4 ∂ B ( i , d ) is cs i -neighborly when i ≤ d − i − 2; 5 B ( i , d ) admits a vertex-transitive action by a group of order 6 4 d (either D 4 d or Z / (2) × D 2 d ). Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction The Construction C ∗ d = cross polytope on { x 1 , . . . , x d , y 1 , . . . , y d } Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction The Construction C ∗ d = cross polytope on { x 1 , . . . , x d , y 1 , . . . , y d } Facets of C ∗ d can be encoded as words in { x , y } : Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction The Construction C ∗ d = cross polytope on { x 1 , . . . , x d , y 1 , . . . , y d } Facets of C ∗ d can be encoded as words in { x , y } : τ = { x 1 , x 2 , y 3 , y 4 , x 5 , y 6 } w ( τ ) = xxyyxy � Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction The Construction C ∗ d = cross polytope on { x 1 , . . . , x d , y 1 , . . . , y d } Facets of C ∗ d can be encoded as words in { x , y } : τ = { x 1 , x 2 , y 3 , y 4 , x 5 , y 6 } w ( τ ) = xxyyxy � The switch set of an xy -word w = w 1 , . . . , w d is S ( w ) := { 1 ≤ j ≤ d − 1 : w j � = w j +1 } . Steven Klee CS Manifolds with Few Vertices

Background Our Result A New Construction The Construction The Construction C ∗ d = cross polytope on { x 1 , . . . , x d , y 1 , . . . , y d } Facets of C ∗ d can be encoded as words in { x , y } : τ = { x 1 , x 2 , y 3 , y 4 , x 5 , y 6 } w ( τ ) = xxyyxy � The switch set of an xy -word w = w 1 , . . . , w d is S ( w ) := { 1 ≤ j ≤ d − 1 : w j � = w j +1 } . w = xxyyxy Steven Klee CS Manifolds with Few Vertices