Balanced shellings on combinatorial manifolds Martina - - PowerPoint PPT Presentation

Balanced shellings on combinatorial manifolds

Martina Juhnke-Kubitzke (joint work with Lorenzo Venturello) Einstein Workshop on Discrete Geometry and Topology 2018, Berlin March 14, 2018

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 1 / 21

1

Balanced combinatorial manifolds

2

Moves on simplicial complexes

3

A balanced analog of Pachner’s theorem for manifolds with boundary

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 2 / 21

Combinatorial manifolds

∆ connected simplicial complex of dimension d.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 3 / 21

Combinatorial manifolds

∆ connected simplicial complex of dimension d. ∆ is a combinatorial d-sphere if it is PL homeomorphic to the boundary of the (d + 1)-simplex.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 3 / 21

Combinatorial manifolds

∆ connected simplicial complex of dimension d. ∆ is a combinatorial d-sphere if it is PL homeomorphic to the boundary of the (d + 1)-simplex. ∆ is a combinatorial d-manifold without boundary if all its vertex links are combinatorial (d − 1)-spheres. The link of F ∈ ∆ is lk∆(F) := {G ∈ ∆ : G ∪ F ∈ ∆, G ∩ F = ∅}.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 3 / 21

Combinatorial manifolds

∆ connected simplicial complex of dimension d. ∆ is a combinatorial d-sphere/d-ball if it is PL homeomorphic to the boundary of the (d + 1)-simplex/d-simplex. ∆ is a combinatorial d-manifold without boundary if all its vertex links are combinatorial (d − 1)-spheres. The link of F ∈ ∆ is lk∆(F) := {G ∈ ∆ : G ∪ F ∈ ∆, G ∩ F = ∅}.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 3 / 21

Combinatorial manifolds

∆ connected simplicial complex of dimension d. ∆ is a combinatorial d-sphere/d-ball if it is PL homeomorphic to the boundary of the (d + 1)-simplex/d-simplex. ∆ is a combinatorial d-manifold without boundary if all its vertex links are combinatorial (d − 1)-spheres. ∆ is a combinatorial d-manifold with boundary if all its vertex links are combinatorial (d − 1)-spheres or (d − 1)-balls and its boundary is ∂∆ := {F ∈ ∆ : lk∆(F) is a combinatorial (d − |F|)-ball} ∪ {∅}. The link of F ∈ ∆ is lk∆(F) := {G ∈ ∆ : G ∪ F ∈ ∆, G ∩ F = ∅}.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 3 / 21

Balanced simplicial complexes

A simplicial complex ∆ on vertex set V (∆) is properly m-colorable, if

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 4 / 21

Balanced simplicial complexes

A simplicial complex ∆ on vertex set V (∆) is properly m-colorable, if the 1-skeleton of ∆ is m-colorable.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 4 / 21

Balanced simplicial complexes

A simplicial complex ∆ on vertex set V (∆) is properly m-colorable, if the 1-skeleton of ∆ is m-colorable. ⇔ There exists a map (coloring) φ : V (∆) → {0, 1, . . . , m − 1}, such that φ(i) = φ(j) for all {i, j} ∈ ∆.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 4 / 21

Balanced simplicial complexes

A simplicial complex ∆ on vertex set V (∆) is properly m-colorable, if the 1-skeleton of ∆ is m-colorable. ⇔ There exists a map (coloring) φ : V (∆) → {0, 1, . . . , m − 1}, such that φ(i) = φ(j) for all {i, j} ∈ ∆. ∆ is balanced if it is properly (dim ∆ + 1)-colorable.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 4 / 21

The (boundary) of the d-simplex

Let σd be the d-simplex. 1 2 3 4

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 5 / 21

The (boundary) of the d-simplex

Let σd be the d-simplex. As the 1-skeleton of σd is a complete graph on d + 1 vertices, a proper coloring uses at least d + 1 colors. 1 2 3 4

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 5 / 21

The (boundary) of the d-simplex

Let σd be the d-simplex. As the 1-skeleton of σd is a complete graph on d + 1 vertices, a proper coloring uses at least d + 1 colors. ⇒ σd is balanced, whereas its boundary ∂σd is not balanced. 1 2 3 4

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 5 / 21

The boundary of the (d + 1)-dimensional cross-polytope

Let Cd be the boundary of the (d + 1)-dimensional cross-polytope: Cd = {v0, w0} ∗ · · · ∗ {vd, wd}. w0 v0 w1 v1 w2 v2

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 6 / 21

The boundary of the (d + 1)-dimensional cross-polytope

Let Cd be the boundary of the (d + 1)-dimensional cross-polytope: Cd = {v0, w0} ∗ · · · ∗ {vd, wd}. A (d + 1)-coloring φ is given by setting φ(vi) = φ(wi) = i for 0 ≤ i ≤ d. w0 v0 w1 v1 w2 v2

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 6 / 21

The boundary of the (d + 1)-dimensional cross-polytope

Let Cd be the boundary of the (d + 1)-dimensional cross-polytope: Cd = {v0, w0} ∗ · · · ∗ {vd, wd}. A (d + 1)-coloring φ is given by setting φ(vi) = φ(wi) = i for 0 ≤ i ≤ d. w0 v0 w1 v1 w2 v2 ⇒ Cd is balanced.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 6 / 21

1

Balanced combinatorial manifolds

2

Moves on simplicial complexes

3

A balanced analog of Pachner’s theorem for manifolds with boundary

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 7 / 21

Stellar moves and bistellar moves

∆ d-dimensional simplicial complex. The stellar subdivision of ∆ at F ∈ ∆ is sdF(∆) = (∆ \ F) ∪ (v ∗ ∂F ∗ lk∆(F)).

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 8 / 21

Stellar moves and bistellar moves

∆ d-dimensional simplicial complex. The stellar subdivision of ∆ at F ∈ ∆ is sdF(∆) = (∆ \ F) ∪ (v ∗ ∂F ∗ lk∆(F)). A bistellar move replaces an induced subcomplex A ⊆ ∆ that is isomorphic to a d-dimensional subcomplex of ∂σd+1 with its complement: ∆ → (∆ \ A) ∪ (∂σd+1 \ A).

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 8 / 21

What about combinatorial manifolds with boundary?

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 9 / 21

Shellings and their inverses

∆ pure d-dimensional simplicial complex. An elementary shelling removes a facet F ∈ ∆ with the property that {G ⊆ F : G / ∈ ∆ \ F} has a unique minimal element. ∆ → ∆ \ F.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 10 / 21

Shellings and their inverses

∆ pure d-dimensional simplicial complex. An elementary shelling removes a facet F ∈ ∆ with the property that {G ⊆ F : G / ∈ ∆ \ F} has a unique minimal element. ∆ → ∆ \ F. The inverse operation is called an inverse shelling.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 10 / 21

Shellings and their inverses

∆ pure d-dimensional simplicial complex. An elementary shelling removes a facet F ∈ ∆ with the property that {G ⊆ F : G / ∈ ∆ \ F} has a unique minimal element. ∆ → ∆ \ F. The inverse operation is called an inverse shelling. A shelling on ∆ corresponds to a bistellar flip on ∂∆.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 10 / 21

What about balanced combinatorial manifolds?

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 11 / 21

Cross-flips

∆ balanced d-dimensional simplicial complex. A cross-flip replaces an induced subcomplex D ⊆ ∆ that is isomorphic to a shellable and coshellable subcomplex of Cd with its complement: ∆ → (∆ \ D) ∪ (Cd \ D).

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 12 / 21

Cross-flips

∆ balanced d-dimensional simplicial complex. A cross-flip replaces an induced subcomplex D ⊆ ∆ that is isomorphic to a shellable and coshellable subcomplex of Cd with its complement: ∆ → (∆ \ D) ∪ (Cd \ D). Cross-flips preserve balancedness. Cross-flips preserve the PL homeomorphism type.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 12 / 21

Cross-flips in dimension 2

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 13 / 21

What about balanced combinatorial manifolds with boundary?

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 14 / 21

1

Balanced combinatorial manifolds

2

Moves on simplicial complexes

3

A balanced analog of Pachner’s theorem for manifolds with boundary

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 15 / 21

The main result

Theorem (J.-K., Venturello; 2018+) Balanced combinatorial manifolds with boundary are PL homeo- morphic if and only if they are connected by a sequence of shellings and inverse shellings preserving balancedness in each step.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 16 / 21

Sketch of the proof

Let ∆ and Γ balanced PL homeomorphic manifolds with boundary.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 17 / 21

Sketch of the proof

Let ∆ and Γ balanced PL homeomorphic manifolds with boundary. Step 1: Convert ∆ via shellings and inverses into a balanced manifold ∆′ such that ∆′ and Γ have isomorphic boundaries.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 17 / 21

Sketch of the proof

Let ∆ and Γ balanced PL homeomorphic manifolds with boundary. Step 1: Convert ∆ via shellings and inverses into a balanced manifold ∆′ such that ∆′ and Γ have isomorphic boundaries. ⇒ ∆′ and Γ are connected by a sequence of bistellar flips.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 17 / 21

Sketch of the proof

Let ∆ and Γ balanced PL homeomorphic manifolds with boundary. Step 1: Convert ∆ via shellings and inverses into a balanced manifold ∆′ such that ∆′ and Γ have isomorphic boundaries. ⇒ ∆′ and Γ are connected by a sequence of bistellar flips. Step 2: Convert the sequence of bistellar flips into a shellable pseudo- cobordism between ∆′ and Γ.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 17 / 21

Sketch of the proof

Let ∆ and Γ balanced PL homeomorphic manifolds with boundary. Step 1: Convert ∆ via shellings and inverses into a balanced manifold ∆′ such that ∆′ and Γ have isomorphic boundaries. ⇒ ∆′ and Γ are connected by a sequence of bistellar flips. Step 2: Convert the sequence of bistellar flips into a shellable pseudo- cobordism between ∆′ and Γ. Step 3: The shellable pseudo-cobordism encodes a sequence of cross-flips ∆′ and Γ.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 17 / 21

Sketch of the proof

Let ∆ and Γ balanced PL homeomorphic manifolds with boundary. Step 1: Convert ∆ via shellings and inverses into a balanced manifold ∆′ such that ∆′ and Γ have isomorphic boundaries. ⇒ ∆′ and Γ are connected by a sequence of bistellar flips. Step 2: Convert the sequence of bistellar flips into a shellable pseudo- cobordism between ∆′ and Γ. Step 3: The shellable pseudo-cobordism encodes a sequence of cross-flips ∆′ and Γ. Step 4: Convert each cross-flip into a sequence of shellings and balanced inverse shellings.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 17 / 21

Step 3: From pseudo-cobordisms to cross-flips

1 2 3 4 5 6 7 8 9 10 ∆1 ∆1 \ ♦(A3) ∪ C2 \ (♦(A3)) Ω ♦(Ω) ♦(F3) F3 = A3 ∗ R3 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 18 / 21

Step 4: From cross-flips to shellings

∆ ∆′ Γ′ Γ F0 F1 F0 F1 G G

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 19 / 21

What else?

We can show that 2d cross-flips suffice to relate any two balanced triangulations of the same manifold. But in dimension 2, Murai and Suzuki showed that 3 = 4 flips suffice and 2 of those are missing from

• ur 4.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 20 / 21

What else?

We can show that 2d cross-flips suffice to relate any two balanced triangulations of the same manifold. But in dimension 2, Murai and Suzuki showed that 3 = 4 flips suffice and 2 of those are missing from

• ur 4.

Lorenzo wrote a program in Sage to apply cross-flips.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 20 / 21

What else?

We can show that 2d cross-flips suffice to relate any two balanced triangulations of the same manifold. But in dimension 2, Murai and Suzuki showed that 3 = 4 flips suffice and 2 of those are missing from

• ur 4.

Lorenzo wrote a program in Sage to apply cross-flips. He found balanced vertex-minimal triangulations of several surfaces and 3-manifolds.

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 20 / 21

Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 21 / 21

Reductions

♦3(Γ1) = ♦3(Γ2)#♦2(Γ1)♦3(Γ2) ♦3(Γ2)#♦2(Γ1)♦3(Γ0)#♦2(Γ{0,1})♦3(Γ{1,2}) ♦3(Γ2)#♦2(Γ1)♦3(Γ{0,1,2}) ♦3(Γ{0,2})#♦2(Γ{0,1})♦3(Γ{1,2}) ♦3(Γ{0,1}) = ♦3(Γ{1,2})#♦2(Γ{0,1})♦3(Γ{1,2})

2 v2 2 v2