Triangulated Surfaces and Higher-Dimensional Manifolds. Frank H. Lutz. (TU Berlin)

Transcription

1 Triangulated Surfaces and Higher-Dimensional Manifolds Frank H. Lutz (TU Berlin)

2 Topology: orientable, genus g = 3 Combinatorics: Geometry: vertex-minimal, n = 10 vertices, irreducible coordinate-minimal, general position

3 Combinatorics: triangulated 2-manifold M : f-vector f = (n, f 1, f 2 ). double counting 3f 2 = 2f 1 Euler characteristic χ(m) = n f 1 + f 2 = f = (n, 3n 3χ(M), 2n 2χ(M))

4 Combinatorics: triangulated 2-manifold M : f-vector f = (n, f 1, f 2 ). double counting 3f 2 = 2f 1 Euler characteristic χ(m) = n f 1 + f 2 = f = (n, 3n 3χ(M), 2n 2χ(M)) number of edges: f 1 ( ) n 2 i.e. n 2 7n + 6χ(M) 0

5 PROPOSITION HEAWOOD (1890) n 1 2 ( χ(M))

6 PROPOSITION HEAWOOD (1890) n 1 2 ( χ(M)) THEOREM RINGEL (1955)/JUNGERMAN & RINGEL (1980) For every surface M, with the exception of M(2, +), M(2, ), and M(3, ), there is a triangulation with n vertices iff n 1 2 ( χ(M)). (In the exceptional cases one vertex has to be added.)

7 Examples: χ(s 2 ) = 2: n 4 χ(rp 2 ) = 1: n 6 χ(t 2 ) = 0: n

8 Enumeration/Realization: [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 4, 5 ], [ 1, 5, 6 ], [ 1, 6, 7 ], [ 2, 3, 6 ], [ 2, 4, 7 ], [ 2, 5, 6 ], [ 2, 5, 7 ], [ 3, 4, 5 ], [ 3, 4, 6 ], [ 3, 5, 7 ], [ 4, 6, 7 ] Möbius torus (1861)

9 Enumeration/Realization: 7 Coordinates: [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 7 ], [ 1, 4, 5 ], [ 1, 5, 6 ], [ 1, 6, 7 ], [ 2, 3, 6 ], [ 2, 4, 7 ], [ 2, 5, 6 ], [ 2, 5, 7 ], [ 3, 4, 5 ], [ 3, 4, 6 ], [ 3, 5, 7 ], [ 4, 6, 7 ] 1: (3,-3,0) 2: (-3,3,0) 3: (-3,-3,1) 4: (3,3,1) 5: (-1,-2,3) 6: (1,2,3) 7: (0,0,15) Császár s torus (1949)

10 Enumeration schemes: 1. Generation from irreducible triangulations. 2. Lexicographic enumeration. 3. Strongly connected enumeration.

11 Enumeration schemes: 1. Generation from irreducible triangulations. 2. Lexicographic enumeration. 3. Strongly connected enumeration.

12 [Royle, 2001]: n Types Triangulated 2-spheres with 4 n 23 vertices

13 edge contraction vertex split A triangulation is irreducible if there is no contractible edge.

14 THEOREM BARNETTE & EDELSON (1989) All 2-manifolds have finitely many irreducible triangulations.

15 THEOREM BARNETTE & EDELSON (1989) All 2-manifolds have finitely many irreducible triangulations. THEOREM NAKAMOTO & OTA (1995) Irreducible triangulations of a closed surface of genus g have at most O(g) vertices.

16 Numbers of irreducible triangulations: 2-sphere: 1 [Steinitz, 1922] 2-torus: 21 [Grünbaum, 1970, Lavrenchenko, 1984] M(2, +): [Sulanke, 2005] M(3, +):? RP 2 2 [Barnette, 1982] Klein bottle: 29 [Sulanke, 2004] M(3, ): [Sulanke, 2005] M(4, ): [Sulanke, 2005] M(5, ):?

17 Enumeration schemes: 1. Generation from irreducible triangulations. 2. Lexicographic enumeration. 3. Strongly connected enumeration.

18 Basic property: I : Every edge lies in exactly two triangles.

19 Ex.: n = 5 triangle-edge incidence matrix

20 Ex.: n = 5 triangle-edge incidence matrix

21 Ex.: n =

22 Ex.: n = 5 backtracking!

23 Ex.: n = 5 backtracking!

24 Ex.: n = 5 backtracking!

25 Ex.: n = 5 backtracking!

26 Ex.: n = 5 backtracking! lexicographic enumeration!

27 Pseudomanifold with isolated singularities:

28 Pseudomanifold with isolated singularities:

29 Pseudomanifold with isolated singularities: II : The link of a vertex should be one circle.

30 Several components:

31 Several components: III : The surface should be connected.

32 Triangulated surfaces with up to 10 vertices [L. 2003]: n Surface Types n Surface Types n Surface Types 4 S S S T 2 7 T S 2 1 M(2, +) 865 RP 2 16 M(3, +) 20 6 S 2 2 K 2 6 RP RP S 2 50 K T M(3, ) S 2 5 M(4, ) T 2 1 RP M(5, ) 7050 K M(6, ) 1022 RP 2 3 M(3, ) 133 M(7, ) 14 M(4, ) 37 M(5, ) 2

33 Isomorphism free lexicographic enumeration: Triangulated surfaces with 11 and 12 vertices [Sulanke & L., 2006].

34 Realization: THEOREM STEINITZ (1922) Every triangulated 2-sphere is realizable as the boundary of a simplicial 3-polytope. CONJECTURE DUKE/GRÜNBAUM (1970/1973) Every triangulated 2-torus is realizable in R 3. THEOREM BOKOWSKI & GUEDES DE OLIVEIRA (2000) There is a non-realizable 12-vertex triangulation of the orientable surface of genus

35 Realization: THEOREM STEINITZ (1922) Every triangulated 2-sphere is realizable as the boundary of a simplicial 3-polytope. CONJECTURE DUKE/GRÜNBAUM (1970/1973) Every triangulated 2-torus is realizable in R 3. THEOREM SCHEWE (2006) For every surface of genus g 5 there is a non-realizable triangulation

36 Realization of vertex-minimal triangulations: g n min Types Realizable? yes yes [Császár, 1949] [L., 2003] yes [Bokowski & L., 2005] [L., 2003] yes [Hougardy, L., Zelke, 2006] [L., 2005] yes [Hougardy, L., Zelke, 2006] [Sulanke, 2005] yes/no [Hougardy, L., Zelke, 2006], [Schewe, 2006] [Bokowski, 1996] no [Bokowski & Guedes de Oliveira, 2000], [Schewe, 2006]

37 Realization heuristics:

38 Realization heuristics: random realization and recycling of coordinates: 864 L. (2005)

39 Realization heuristics: random realization and recycling of coordinates: 864 L. (2005) geometric intuition: 1 Bokowski (2005)

40 small coordinates: Hougardy, L., Zelke (2005)

41 THEOREM HOUGARDY, L., ZELKE (2005) All 865 triangulations of the orientable surface of genus 2 with 10 vertices are realizable in the (4 4 4)-cube.

42 THEOREM HOUGARDY, L., ZELKE (2006) At least 17 of the 20 triangulations of the orientable surface of genus 3 with 10 vertices are realizable in the (5 5 5)-cube.

43 THEOREM HOUGARDY, L., ZELKE (2006) At least 17 of the 20 triangulations of the orientable surface of genus 3 with 10 vertices are realizable in the (5 5 5)-cube. THEOREM HOUGARDY, L., ZELKE (2005) All 20 triangulations of the orientable surface of genus 3 with 10 vertices are realizable.

44 intersection edge functional: Hougardy, L., Zelke (2005) E f a c u w b v d e

45 Example:

46 g = 4, n = 11: All 821 examples are realizable [HLZ, 2006]. g = 5, n = 12: At least 15 of the examples are realizable [HLZ, 2006] and at least 3 are not realizable [Schewe, 2006].

47 g = 4, n = 11: All 821 examples are realizable [HLZ, 2006]. g = 5, n = 12: At least 15 of the examples are realizable [HLZ, 2006] and at least 3 are not realizable [Schewe, 2006]. CONJECTURE Every triangulated surface of genus 1 g 4 is realizable in R

48 Enumeration schemes: 1. Generation from irreducible triangulations. 2. Lexicographic enumeration. 3. Strongly connected enumeration.

49 Date: Thu, 3 Feb :23: From: John M Sullivan <Sullivan@math.TU-Berlin.DE> Subject: triangulations with low edge degree Dear Frank,... Consider triangulated 3-manifolds with every edge valence at most 5. Are there really (as I suspect) only a finite number of such triangulations, and are the manifolds then all spherical?? Do you perhaps already have such a list? If not, can we modify your programs to produce one? Best, John

50 Date: Thu, 3 Feb :23: From: John M Sullivan <Sullivan@math.TU-Berlin.DE> Subject: triangulations with low edge degree Dear Frank,... Consider triangulated 3-manifolds with every edge valence at most 5. Are there really (as I suspect) only a finite number of such triangulations, and are the manifolds then all spherical? Do you perhaps already have such a list? If not, can we modify your programs to produce one?? No :-( Best, John

51 Date: Thu, 3 Feb :23: From: John M Sullivan <Sullivan@math.TU-Berlin.DE> Subject: triangulations with low edge degree Dear Frank,... Consider triangulated 3-manifolds with every edge valence at most 5. Are there really (as I suspect) only a finite number of such triangulations, and are the manifolds then all spherical? Do you perhaps already have such a list? If not, can we modify your programs to produce one? Best, John? No :-( Yes :-)

52 Vertex-links: 2-spheres with face vector f = (n, f 1, f 2 )

53 Vertex-links: 2-spheres with face vector f = (n, f 1, f 2 ). By Euler s equation n f 1 + f 2 = 2 and double counting 2f 1 = 3f 2 = f = (n, 3n 6, 2n 4)

54 Vertex-links: 2-spheres with face vector f = (n, f 1, f 2 ). By Euler s equation n f 1 + f 2 = 2 and double counting 2f 1 = 3f 2 = f = (n, 3n 6, 2n 4) Vertex-degree 5: 2f 1 = deg 5n. Thus n 12

55 Enumeration (of vertex-links):

56 Enumeration (of vertex-links):

57 Enumeration (of vertex-links):

58 Enumeration (of vertex-links):

59 Enumeration (of vertex-links):......

60 Enumeration (of vertex-links):......

61 Enumeration (of vertex-links):

62 Enumeration (of vertex-links):

63 Enumeration (of vertex-links):

64 Enumeration (of vertex-links): strongly connected!......

65 Vertex-links:

66 Combinatorial types of 3-manifolds with edge valence at most 5: n Types n Types n Types n Types n Types

67 THEOREM L. & SULLIVAN (2005) There are exactly 4787 combinatorially distinct triangulated 3-manifolds with edge valence at most five: 4761 S 3, 22 RP 3, 2 L(3, 1), 1 L(4, 1), 1 S 3 /Q. THEOREM MATVEEV & SHEVCHISHIN (2005) Let M be a triangulated 3-manifold such that every edge has valence at most five. Then M is spherical and has at most 600 tetrahedra.

68 THEOREM L. & SULLIVAN (2006) There are exactly 41 distinct 3-dimensional combinatorial pseudomanifolds with edge valence at most five. They all are of type susp(rp 2 ).

69 THEOREM L. & SULLIVAN (2006) There are exactly 41 distinct 3-dimensional combinatorial pseudomanifolds with edge valence at most five. They all are of type susp(rp 2 ). THEOREM L. & SULLIVAN (2005) Let M be a triangulated d-manifold such that every codimension-two face has valence at most four. Then M is the join product of boundaries of simplices.

70 Recognition: Algorithms for the recognition of the 3-sphere: [Rubinstein, 1992], [Thompson, 1994], [S. Matveev, 1995],...

71 Recognition: Algorithms for the recognition of the 3-sphere: [Rubinstein, 1992], [Thompson, 1994], [S. Matveev, 1995], are exponential, difficult to implement.

72 Recognition: Algorithms for the recognition of the 3-sphere: [Rubinstein, 1992], [Thompson, 1994], [S. Matveev, 1995], are exponential, difficult to implement. Heuristics for the recognition of the 3-sphere: [Björner & L., 1997]... simulated annealing with bistellar flips.

73 Bistellar Flips/Pachner moves! 2d: 3d:

74 THEOREM CSORBA & L. (2005) Hom(C 6, K 5 ) = (S 2 S 2 ) #29. f = (1920, 30780, , , 50400) = f = (33, 379, 1786, 2300, 920) THEOREM CSORBA & L. (2005) Hom(C 5, K 5 ) = V 4,2 = S 2 S 3. f = (1020, 25770, , , , 94400) = f = (12, 66, 220, 390, 336, 112)

75 BUT there is a 3-sphere with no flips (16 vertices) [Dougherty, Faber, and Murphy, 2004]... there are non-shellable/non-constructible 3-spheres with n 13 vertices [L., 2004]... there are worse examples [King, 2004]... there is no algorithm to recognize 4-manifolds [Markov, 1958]... there is no algorithm to recognize d-spheres for d 5 [Novikov, 1962]... there are non-pl spheres [Edwards, 1975], [Cannon, 1979]... there are non-pl d-spheres with d + 13 vertices for d 5 [Björner & L., 2000]

76 THEOREM KING (2004) From any triangulation of S 3, one can obtain an edge contractible triangulation of S 3 by a sequence of at most 2 401f2 3 successive expansions.

77 THEOREM KING (2004) From any triangulation of S 3, one can obtain an edge contractible triangulation of S 3 by a sequence of at most 2 401f2 3 successive expansions. THEOREM LICKORISH (1991), ZIEGLER (1996), KING (2004) Every 3-manifold has infinitely many irreducible triangulations.

78 Why don t we run into bad examples?

79 Why don t we run into bad examples? Where do we get our examples from?

80 Why don t we run into bad examples? Where do we get our examples from? 1. Enumeration: bad examples need more space. 2. Combinatorial constructions: mostly harmless. 3. Topological constructions: we usually know the outcome in advance.

81 Construction of a non-realizable 3-ball with 16 vertices:

82 Construction of a non-realizable 3-ball with 16 vertices: non-realizable [non-constructible/non-shellable] 3-ball: (16, 75, 106, 46) [non-constructible/non-shellable 3-sphere: (17, 91, 148, 74)]

83 Construction of a non-realizable 3-ball with 12 vertices: non-realizable [non-constructible/non-shellable] 3-ball: (12, 58, 84, 37) [non-constructible/non-shellable 3-sphere: (13, 69, 112, 56)]

84 COROLLARY L. (2003) There are non-realizable [non-constructible] d-balls with d + 9 vertices and 37 facets for d 3.

85 COROLLARY L. (2003) There are non-realizable [non-constructible] d-balls with d + 9 vertices and 37 facets for d 3. COROLLARY L. (2003) There are non-constructible/non-shellable d-spheres with d + 10 vertices for d 3.