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

1

Background Definitions Products of Spheres Known Results

2

A New Construction Our Result The Construction

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Definitions Products of Spheres 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

Background A New Construction Definitions Products of Spheres Known Results

CS Neighborly Complexes

Example: C∗

d := conv{±e1, ±e2, . . . , ±ed} ⊆ Rd is centrally

symmetric under the involution ϕ(ei) = −ei.

y1 x1 x2 y2 x3 y3 Steven Klee CS Manifolds with Few Vertices

Background A New Construction Definitions Products of Spheres Known Results

CS Neighborly Complexes

Definition A centrally symmetric simplicial complex ∆ on 2d vertices is (cs) k-neighborly if it has the k-skeleton of C∗

d.

Example: A cs 1-neighborly triangulation of S1 × S1.

x1 y3 y1 x3 x1 y4 x2 y4 x2 x1 y3 y1 x3 x1 y2 x4

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Definitions Products of Spheres Known Results

CS Products of Spheres

Proposition A cs triangulation of Si × Sd−i−2 requires at least 2d vertices. Question

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Definitions Products of Spheres Known Results

CS Products of Spheres

Proposition A cs triangulation of Si × Sd−i−2 requires at least 2d vertices. Question

1

Do there exist cs triangulations of Si × Sd−i−2 with exactly 2d vertices?

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Definitions Products of Spheres Known Results

CS Products of Spheres

Proposition A cs triangulation of Si × Sd−i−2 requires at least 2d vertices. Question

1

Do there exist cs triangulations of Si × Sd−i−2 with exactly 2d vertices?

2

(Sparla) Are there i-neighborly triangulations of Si × Si with exactly 4i + 4 vertices?

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Definitions Products of Spheres Known Results

CS Products of Spheres

Proposition A cs triangulation of Si × Sd−i−2 requires at least 2d vertices. Question

1

Do there exist cs triangulations of Si × Sd−i−2 with exactly 2d vertices?

2

(Sparla) Are there i-neighborly triangulations of Si × Si with exactly 4i + 4 vertices?

3

(Lutz) Are there ⌊d

2 ⌋-neighborly triangulations of S⌈ d

2 ⌉ × S⌊ d 2 ⌋

• n 2d + 4 vertices that admit a dihedral group action?

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Definitions Products of Spheres Known Results

Known Results

Question 1: Do there exist cs triangulations of Si × Sd−i−2 with exactly 2d vertices? Theorem (K¨ uhnel-Lassmann) There exists a cs triangulation of S1 × Sd−3 on 2d vertices for all d ≥ 3. (Lutz) Such triangulations exist for all d ≤ 10 (computer check).

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Definitions Products of Spheres Known Results

Known Results

Question 2: (Sparla) Are there i-neighborly triangulations of Si × Si with exactly 4i + 4 vertices? Examples Triangulation of S1 × S1 on 8 vertices. Triangulation of S2 × S2 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

Background A New Construction Definitions Products of Spheres Known Results

Known Results

Question 3: Are there ⌊d

2 ⌋-neighborly triangulations of

S⌈ d

2 ⌉ × S⌊ d 2 ⌋ on 2d + 4 vertices that admit a dihedral group action?

Theorem (Lutz) When d ≤ 8, there are cs triangulations of Si × Sd−i−2 that admit a vertex-transitive action by a dihedral group of

• rder 4d

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Definitions Products of Spheres Known Results

Known Results

Question 3: Are there ⌊d

2 ⌋-neighborly triangulations of

S⌈ d

2 ⌉ × S⌊ d 2 ⌋ on 2d + 4 vertices that admit a dihedral group action?

Theorem (Lutz) When d ≤ 8, there are cs triangulations of Si × Sd−i−2 that admit a vertex-transitive action by a dihedral group of

• rder 4d EXCEPT in the cases of S2 × S4 and S2 × S6.

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result 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:

1

B(i, d) is centrally symmetric on 2d vertices;

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result 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:

1

B(i, d) is centrally symmetric on 2d vertices;

2

B(i, d) is cs i-neighborly;

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result 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:

1

B(i, d) is centrally symmetric on 2d vertices;

2

B(i, d) is cs i-neighborly;

3

• H∗(B(i, d); Z) ≈

H∗(Si; Z);

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result 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:

1

B(i, d) is centrally symmetric on 2d vertices;

2

B(i, d) is cs i-neighborly;

3

• H∗(B(i, d); Z) ≈

H∗(Si; Z);

4

∂B(i, d) triangulates Si × Sd−i−2;

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result 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:

1

B(i, d) is centrally symmetric on 2d vertices;

2

B(i, d) is cs i-neighborly;

3

• H∗(B(i, d); Z) ≈

H∗(Si; Z);

4

∂B(i, d) triangulates Si × Sd−i−2;

5

∂B(i, d) is cs i-neighborly when i ≤ d − i − 2;

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result 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:

1

B(i, d) is centrally symmetric on 2d vertices;

2

B(i, d) is cs i-neighborly;

3

• H∗(B(i, d); Z) ≈

H∗(Si; Z);

4

∂B(i, d) triangulates Si × Sd−i−2;

5

∂B(i, d) is cs i-neighborly when i ≤ d − i − 2;

6

B(i, d) admits a vertex-transitive action by a group of order 4d (either D4d or Z/(2) × D2d).

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

C∗

d = cross polytope on {x1, . . . , xd, y1, . . . , yd}

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

C∗

d = cross polytope on {x1, . . . , xd, y1, . . . , yd}

Facets of C∗

d can be encoded as words in {x, y}:

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

C∗

d = cross polytope on {x1, . . . , xd, y1, . . . , yd}

Facets of C∗

d can be encoded as words in {x, y}:

τ = {x1, x2, y3, y4, x5, y6}

• w(τ) = xxyyxy

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

C∗

d = cross polytope on {x1, . . . , xd, y1, . . . , yd}

Facets of C∗

d can be encoded as words in {x, y}:

τ = {x1, x2, y3, y4, x5, y6}

• w(τ) = xxyyxy

The switch set of an xy-word w = w1, . . . , wd is S(w) := {1 ≤ j ≤ d − 1 : wj = wj+1}.

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

C∗

d = cross polytope on {x1, . . . , xd, y1, . . . , yd}

Facets of C∗

d can be encoded as words in {x, y}:

τ = {x1, x2, y3, y4, x5, y6}

• w(τ) = xxyyxy

The switch set of an xy-word w = w1, . . . , wd is S(w) := {1 ≤ j ≤ d − 1 : wj = wj+1}.

w = xxyyxy

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

C∗

d = cross polytope on {x1, . . . , xd, y1, . . . , yd}

Facets of C∗

d can be encoded as words in {x, y}:

τ = {x1, x2, y3, y4, x5, y6}

• w(τ) = xxyyxy

The switch set of an xy-word w = w1, . . . , wd is S(w) := {1 ≤ j ≤ d : wj = wj+1}.

w = xxyyxy

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

C∗

d = cross polytope on {x1, . . . , xd, y1, . . . , yd}

Facets of C∗

d can be encoded as words in {x, y}:

τ = {x1, x2, y3, y4, x5, y6}

• w(τ) = xxyyxy

The switch set of an xy-word w = w1, . . . , wd is S(w) := {1 ≤ j ≤ d : wj = wj+1}.

w = xxyyxy

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

C∗

d = cross polytope on {x1, . . . , xd, y1, . . . , yd}

Facets of C∗

d can be encoded as words in {x, y}:

τ = {x1, x2, y3, y4, x5, y6}

• w(τ) = xxyyxy

The switch set of an xy-word w = w1, . . . , wd is S(w) := {1 ≤ j ≤ d : wj = wj+1}.

w = xxyyxy

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

C∗

d = cross polytope on {x1, . . . , xd, y1, . . . , yd}

Facets of C∗

d can be encoded as words in {x, y}:

τ = {x1, x2, y3, y4, x5, y6}

• w(τ) = xxyyxy

The switch set of an xy-word w = w1, . . . , wd is S(w) := {1 ≤ j ≤ d : wj = wj+1}.

w = xxyyxy

• S(w) = {2, 4, 5}

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

The switch set of an xy-word w = w1, . . . , wd is S(w) := {1 ≤ j ≤ d : wj = wj+1}.

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

The Construction

The switch set of an xy-word w = w1, . . . , wd is S(w) := {1 ≤ j ≤ d : wj = wj+1}. Definition The complex B(i, d) is the subcomplex of C∗

d induced by all facets

τ with |S(τ)| ≤ i.

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

An Example

Definition The complex B(i, d) is the subcomplex of C∗

d induced by all facets

τ with |S(τ)| ≤ i. Example: B(1, 3)

x1 x3 y2 x1 x2 y1 y3 x2 S(τ) τx τy ∅ x1x2x3 y1y2y3 {1} y1x2x3 x1y2y3 {2} x1x2y3 y1y2x3

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

Other Interesting Properties

Let C(i, d) denote the subcomplex of C∗

d induced by the facets not

contained in B(i, d).

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

Other Interesting Properties

Let C(i, d) denote the subcomplex of C∗

d induced by the facets not

contained in B(i, d). Theorem C(i, d) is combinatorially isomorphic to B(d − i − 2, d) C(i, d) intersects B(i, d) along their common boundary

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

Other Interesting Properties

Let C(i, d) denote the subcomplex of C∗

d induced by the facets not

contained in B(i, d). Theorem C(i, d) is combinatorially isomorphic to B(d − i − 2, d) C(i, d) intersects B(i, d) along their common boundary Example B(1, 4) is a solid torus; C(1, 4) ≈ B(1, 4) is also a solid torus.

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

Other Interesting Properties

Let C(i, d) denote the subcomplex of C∗

d induced by the facets not

contained in B(i, d). Theorem C(i, d) is combinatorially isomorphic to B(d − i − 2, d) C(i, d) intersects B(i, d) along their common boundary Example B(1, 4) is a solid torus; C(1, 4) ≈ B(1, 4) is also a solid torus. This gives the standard decomposition of S3 = C∗

4 as the

union of two tori intersecting along their boundary.

Steven Klee CS Manifolds with Few Vertices

Background A New Construction Our Result The Construction

Why does ∂B(i, d) triangulate Si × Sd−i−2?

Theorem (Kreck) If M ⊂ Sd−1 is a simply connected (d − 2)-submanifold with H∗(M; Z) ≈ H∗(Si × Sd−i−2; Z), and d ≥ 6, then M ≈ Si × Sd−i−2. Question Does B(i, d) triangulate Si × Bd−i−1?

Steven Klee CS Manifolds with Few Vertices