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Jun 13, 2023 12 likes •416 views

Geometric bistellar moves relate triangulations of Euclidean, hyperbolic and spherical manifolds Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) 17th March, 2020 Tejas Kalelkar, Indian

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Geometric bistellar moves relate triangulations of Euclidean, hyperbolic and spherical manifolds

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) 17th March, 2020

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric Flips

Figure: Flips relate any two triangulations of a 2-polytope with same vertices.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric Flips

Figure: Flips relate any two triangulations of a 2-polytope with same vertices.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric Flips

Figure: Flips relate any two triangulations of a 2-polytope with same vertices.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric Flips

Figure: Flips relate any two triangulations of a 2-polytope with same vertices.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric Flips

Figure: Flips relate any two triangulations of a 2-polytope with same vertices.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric Flips

Figure: Flips relate any two triangulations of a 2-polytope with same vertices.

Theorem (Despre - Schlenker - Teillaud)

Let S be either a torus with a Euclidean metric or a closed oriented surface with a hyperbolic metric. Then any two geometric triangulations of S with the same vertex set are related by geometric flips.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric Flips

Figure: Flips relate any two triangulations of a 2-polytope with same vertices.

Theorem (Despre - Schlenker - Teillaud)

Let S be either a torus with a Euclidean metric or a closed oriented surface with a hyperbolic metric. Then any two geometric triangulations of S with the same vertex set are related by geometric flips.

Theorem (Santos)

There exist 5-dimensional polytopes with triangulations with the same vertex set which are not related by geometric flips.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric Flips

Figure: Flips relate any two triangulations of a 2-polytope with same vertices.

Theorem (Despre - Schlenker - Teillaud)

Let S be either a torus with a Euclidean metric or a closed oriented surface with a hyperbolic metric. Then any two geometric triangulations of S with the same vertex set are related by geometric flips.

Theorem (Santos)

There exist 5-dimensional polytopes with triangulations with the same vertex set which are not related by geometric flips.

Question

When the vertex sets are possibly different, what classes of triangulations are related by n-dimensional geometric bistellar moves?

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric bistellar moves

Definition

Let K be a triangulation of an n-manifold M and let D be a disk-subcomplex

  • f K simplicially isomorphic to an n-disk in ∂∆n+1. Then a bistellar move on D

replaces D with the disk isomorphic to ∂∆n+1 \ int(D).

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric bistellar moves

Definition

Let K be a triangulation of an n-manifold M and let D be a disk-subcomplex

  • f K simplicially isomorphic to an n-disk in ∂∆n+1. Then a bistellar move on D

replaces D with the disk isomorphic to ∂∆n+1 \ int(D).

Figure: A 2-2 bistellar move

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric bistellar moves

Definition

Let K be a triangulation of an n-manifold M and let D be a disk-subcomplex

  • f K simplicially isomorphic to an n-disk in ∂∆n+1. Then a bistellar move on D

replaces D with the disk isomorphic to ∂∆n+1 \ int(D).

Figure: A 2-2 bistellar move Figure: A 3-1 and 1-3 bistellar move

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric bistellar moves

Definition

Let K be a triangulation of an n-manifold M and let D be a disk-subcomplex

  • f K simplicially isomorphic to an n-disk in ∂∆n+1. Then a bistellar move on D

replaces D with the disk isomorphic to ∂∆n+1 \ int(D).

Figure: A 1-4 and 4-1 bistellar move

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric bistellar moves

Definition

Let K be a triangulation of an n-manifold M and let D be a disk-subcomplex

  • f K simplicially isomorphic to an n-disk in ∂∆n+1. Then a bistellar move on D

replaces D with the disk isomorphic to ∂∆n+1 \ int(D).

Figure: A 1-4 and 4-1 bistellar move Figure: A 2-3 and 3-2 bistellar move

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric bistellar moves relate triangulations

Question

When the vertex sets are possibly different, what classes of triangulations are related by n-dimensional geometric bistellar moves?

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric bistellar moves relate triangulations

Question

When the vertex sets are possibly different, what classes of triangulations are related by n-dimensional geometric bistellar moves?

Theorem (Izmestiev - Schlenker)

Any two triangulations of a convex polytope in R3 can be connected by a sequence of geometric bistellar moves, boundary geometric stellar moves and continuous displacements of the interior vertices.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Geometric bistellar moves relate triangulations

Question

When the vertex sets are possibly different, what classes of triangulations are related by n-dimensional geometric bistellar moves?

Theorem (Izmestiev - Schlenker)

Any two triangulations of a convex polytope in R3 can be connected by a sequence of geometric bistellar moves, boundary geometric stellar moves and continuous displacements of the interior vertices.

Theorem

Let K1 and K2 be geometric simplicial triangulations (with possibly different vertex sets) of a compact Euclidean, hyperbolic or spherical n-manifold M. If M is spherical, we assume that the star of each simplex has diameter less than π. Let L be a possibly empty common subcomplex of K1 and K2. If M has boundary then we insist that K1 and K2 agree on ∂M, i.e., |L| ⊃ ∂M. When n is 2 or 3, then K1 and K2 are related by geometric bistellar moves which keep L fixed. When n > 3, then some s-th iterated derived subdivisions βsK1 and βsK2 are related by geometric bistellar moves which keep βsL fixed.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Simplicial cobordism

K L

Figure: Two triangulations K and L of a hyperbolic manifold M

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Simplicial cobordism

K L

Figure: A geometric triangulation of M × I from K to L

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Simplicial cobordism

K1 L

Figure: Removing an n-simplex from the top and then projecting the upper boundary down to M × 0 gives a bistellar move from K to K1

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Simplicial cobordism

K2 L

Figure: Removing an n-simplex from the top and then projecting the upper boundary down to M × 0 gives a bistellar move from K1 to K2

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Simplicial cobordism

K3 L

Figure: Removing an n-simplex from the top and then projecting the upper boundary down to M × 0 gives a bistellar move from K2 to K3

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Simplicial cobordism

K4 L

Figure: Removing an n-simplex from the top and then projecting the upper boundary down to M × 0 gives a bistellar move from K3 to K4

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Simplicial cobordism

Question

Is there a geometric triangulation of M × I in Hn × R geometry, with the given triangulations K and L on M × 0 and M × 1?

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Simplicial cobordism

Question

Is there a geometric triangulation of M × I in Hn × R geometry, with the given triangulations K and L on M × 0 and M × 1?

Theorem (Cartan)

If at every point p ∈ M and for every subspace V of TpM there exists a totally geodesic surface S through p with TpS = V then M has constant curvature.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Simplicial cobordism

Question

Is there a geometric triangulation of M × I in Hn × R geometry, with the given triangulations K and L on M × 0 and M × 1?

Theorem (Cartan)

If at every point p ∈ M and for every subspace V of TpM there exists a totally geodesic surface S through p with TpS = V then M has constant curvature.

Question

Is it possible to get an enumeration ∆0, ∆1, ..., ∆m of the n-simplexes such that the projection pr : ∪m

j=i∆j → M × 0 is an injection when restricted to the

upper boundary of each ∪m

j=i∆j?

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Simplicial cobordism

Question

Is there a geometric triangulation of M × I in Hn × R geometry, with the given triangulations K and L on M × 0 and M × 1?

Theorem (Cartan)

If at every point p ∈ M and for every subspace V of TpM there exists a totally geodesic surface S through p with TpS = V then M has constant curvature.

Question

Is it possible to get an enumeration ∆0, ∆1, ..., ∆m of the n-simplexes such that the projection pr : ∪m

j=i∆j → M × 0 is an injection when restricted to the

upper boundary of each ∪m

j=i∆j?

K L

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Common subdivision

Let K ∗ = β(K1 ∩ K2) be a common geometric subdivision of K1 and K2. Any constant curvature manifold M has local maps taking balls in M to En by a homeomorphism taking geodesics to straight lines. So stars of simplexes in K ∗ are identified with star-convex n-polytopes in En.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Common subdivision

Let K ∗ = β(K1 ∩ K2) be a common geometric subdivision of K1 and K2. Any constant curvature manifold M has local maps taking balls in M to En by a homeomorphism taking geodesics to straight lines. So stars of simplexes in K ∗ are identified with star-convex n-polytopes in En. We show that K ∗ is related to βK1 and to βK2 by geometric bistellar moves that change the star of each simplex to the cone over its boundary.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Common subdivision

Let K ∗ = β(K1 ∩ K2) be a common geometric subdivision of K1 and K2. Any constant curvature manifold M has local maps taking balls in M to En by a homeomorphism taking geodesics to straight lines. So stars of simplexes in K ∗ are identified with star-convex n-polytopes in En. We show that K ∗ is related to βK1 and to βK2 by geometric bistellar moves that change the star of each simplex to the cone over its boundary. In dimension 2 and 3, K ∼ βKi by geometric bistellar moves.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Outline of proof

Figure: A simplical complex K and its subdivision K ∗

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Outline of proof

Figure: Complex K ′ obtained by replacing Star(A, K) with C(∂Star(A, K)), for A varying over 2-dimensional simplexes

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Outline of proof

Figure: Complex βK obtained by replacing Star(A, K ′) with C(∂Star(A, K)), for A varying over 1-dimensional simplexes

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Outline of proof

Figure: Complex βK is obtained from K ∗ by replacing Star(A, K) with C(∂Star(A, K)) inductively over dimension of A

Enough to show that star-convex polytopes in En can be starred, i.e., any linear triangulation of a star-convex polytope can be changed to a cone

  • ver it’s boundary by Euclidean bistellar moves. Then βK1 ∼ K ∗ ∼ βK2.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Outline of Proof

K C(∂K)

Figure: Cone over a star-convex n-polytope K

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Outline of Proof

K C(∂K)

Figure: Cone over a star-convex n-polytope K

We call a triangulation K of a polytope P regular if there is a function h : |K| → R that is linear on each simplex of K and strictly convex across codimension one simplexes of K, i.e., if points x and y are in neighboring top-dimensional simplexes of K then the segment connecting h(x) and h(y) is above the graph of h (except at the end points).

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Outline of Proof

K C(∂K)

Figure: Cone over a star-convex n-polytope K

We call a triangulation K of a polytope P regular if there is a function h : |K| → R that is linear on each simplex of K and strictly convex across codimension one simplexes of K, i.e., if points x and y are in neighboring top-dimensional simplexes of K then the segment connecting h(x) and h(y) is above the graph of h (except at the end points). Let K denote a triangulated n-polytope in En. We show that for some s ∈ N the s-th derived subdivision βsK is regular.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Outline of Proof

K C(∂K)

Figure: Cone over a star-convex n-polytope K

We call a triangulation K of a polytope P regular if there is a function h : |K| → R that is linear on each simplex of K and strictly convex across codimension one simplexes of K, i.e., if points x and y are in neighboring top-dimensional simplexes of K then the segment connecting h(x) and h(y) is above the graph of h (except at the end points). Let K denote a triangulated n-polytope in En. We show that for some s ∈ N the s-th derived subdivision βsK is regular. (Assume s = 0 for simplicity) Let h : |K| → R be a regular function. Enumerating n-simplexes in decreasing order of ∂h/∂xn+1, we get the required sequence ∆0, ..., ∆m such that the projection pr : ∪m

j=i∆j → M × 0 is injective on the upper boundary of ∪m j=i∆j and

therefore K ∼ C(∂K) by geometric bistellar moves.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Outline of Proof

K C(∂K)

Figure: Cone over a star-convex n-polytope K

We call a triangulation K of a polytope P regular if there is a function h : |K| → R that is linear on each simplex of K and strictly convex across codimension one simplexes of K, i.e., if points x and y are in neighboring top-dimensional simplexes of K then the segment connecting h(x) and h(y) is above the graph of h (except at the end points). Let K denote a triangulated n-polytope in En. We show that for some s ∈ N the s-th derived subdivision βsK is regular. (Assume s = 0 for simplicity) Let h : |K| → R be a regular function. Enumerating n-simplexes in decreasing order of ∂h/∂xn+1, we get the required sequence ∆0, ..., ∆m such that the projection pr : ∪m

j=i∆j → M × 0 is injective on the upper boundary of ∪m j=i∆j and

therefore K ∼ C(∂K) by geometric bistellar moves. Inductively starring the stars of simplexes in decreasing order of their dimension, we get a sequence of geometric bistellar moves βs+1K1 ∼ βsK ∗ ∼ βs+1K2 as required.

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds

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Thank you

Danke Schoen!

Tejas Kalelkar, Indian Institute of Science Education and Research, Pune (Joint work with Advait Phanse) Geometric bistellar moves on triangulations of constant curvature manifolds