Elements of Combinatorial Topology

Transcription

1 Elements of Combinatorial Topology Companion slides for Distributed Computing Through Maurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum 1

2 Road Map Simplicial Complexes Standard Constructions Carrier Maps Connectivity Subdivisions Simplicial & Continuous Approximations 2

3 Road Map Simplicial Complexes Standard Constructions Carrier Maps Connectivity Subdivisions Simplicial & Continuous Approximations 3

4 A Vertex Combinatorial: an element of a set. Geometric: a point in highdimensional Euclidean Space 4

5 Simplexes Combinatorial: a set of vertexes. Geometric: convex hull of points in general position 0-simplex 1-simplex dimension 2-simplex 3-simplex 5

6 Simplicial Complex Combinatorial: a set of simplexes close Geometric: under inclusion. simplexes glued together along faces 6

7 Graphs vs Complexes dimension 0 or 1 arbitrary dimension complexes are a natural generalization of graphs 7

8 Abstract Simplicial Complex finite set V with a collection K of subsets of V, such that 8

9 Abstract Simplicial Complex finite set V with a collection K of subsets of V, such that 1. for all s 2 S, {s} 2 K 9

10 Abstract Simplicial Complex finite set S with a collection K of subsets of S, such that 1. for all s 2 S, {s} 2 K 2. for all X 2 K, and Y ½ X, Y 2 K 10

11 Geometric Simplicial Complex A collection of geometric simplices in R d such that 11

12 Geometric Simplicial Complex A collection of geometric simplices in R d such that 1. any face of a ¾2K is also in K 12

13 Geometric Simplicial Complex A collection of geometric simplices in R d such that 1. any face of a ¾2K is also in K 2. for all ¾, 2 K, their intersection ¾ Å is a face of each of them. 13

14 Abstract vs Geometric Complexes Abstract: A Geometric: A 14

15 Simplicial Maps A Á B Vertex-to-vertex map 15

16 Simplicial Map A Á B Vertex-to-vertex map that sends simplexes to simplexes Á: A! B 16

17 Road Map Simplicial Complexes Standard Constructions Carrier Maps Connectivity Subdivisions Simplicial & Continuous Approximations 17

18 Skeleton C skel 1 C skel 0 C 18

19 Facet A facet of K is a simplex of maximal dimension 19

20 Star Star(¾,K) is the complex of facets of K containing ¾ Complex 20

21 Open Star Star o (¾,K) union of interiors of simplexes containing ¾ Point Set 21

22 Link Link(¾,K) is the complex of simplices of Star(¾,K) not containing ¾ Complex 22

23 More Links v v C Link(v,C)

24 More Links e e C Link(e,C)

25 Join Let A and B be complexes with disjoint sets of vertices their join A*B is the complex with vertices V(A) [ V(B) and simplices [, where 2 A, and 2 B. 25

26 Join A B A*B 26

27 Road Map Simplicial Complexes Standard Constructions Carrier Maps Connectivity Subdivisions Simplicial & Continuous Approximations 27

28 Carrier Map A B Maps simplex of A to subcomplex of B : A! 2 B 28

29 Carrier Maps are Monotonic A B If µ ¾ then ( ) µ (¾) or for ¾, 2 A, (¾Å ) µ (¾)Å ( ) 29

30 Example

31 Example on vertices

32 Example on edges There is no simplicial map carried by

33 Strict Carrier Maps A B for all ¾, 2 A, (¾Å ) = (¾)Å ( ) replace µ with = 33

34 Rigid Carrier Maps A B for ¾ 2 A, (¾) is pure of dimension dim ¾ 34

35 Carrier of a Simplex A B given strict : A! 2 B for each 2 B, 9 unique smallest ¾ 2 A such that 2 (¾). ¾ = Car(, (¾)) sometimes omitted 35

36 Carrier Map Carried By Carrier Map Given carrier maps : A! 2 B ª: A! 2 B is carried by ª if for all ¾ 2 A, (¾) µ ª(¾) written: µ ª 36

37 Simplicial Map Carried By Carrier Map Given carrier and simplicial maps : A! 2 B ϕ: A! B ϕ is carried by if for all ¾ 2 A, ϕ(¾) µ (¾) written: ϕ µ 37

38 Continuous Map Carried By Carrier Map Given carrier and continuous maps : A! 2 B f: A! B f is carried by if for all ¾ 2 A, f(¾) µ (¾) 38

39 Compositions Given carrier maps : A! 2 B ª: B! 2 C their composition is (ª )(¾) := [ ª( ) 2 (¾) 39

40 Theorem If, ª are both strict so is ª rigid so is ª 40

41 Compositions Given carrier and simplicial maps : A! 2 B ϕ: C! A their composition is the carrier map ( ϕ): C! 2 B defined by ( ϕ)(¾) := (ϕ(¾)) 41

42 Compositions Given carrier and simplicial maps : A! 2 B ϕ: B! C their composition is the carrier map (ϕ ): A! 2 C defined by ( ϕ)(¾) := [ 2 (¾) ϕ( ) 42

43 Colorings n := 43

44 Chromatic Complex  A rigid simplicial map n 44

45 Color-Preserving Simplicial Map ϕ A n color of v = color of ϕ(v) 45

46 Road Map Simplicial Complexes Standard Constructions Carrier Maps Connectivity Subdivisions Simplicial & Continuous Approximations 46

47 A Path simplicial complex vertex edge edge vertex vertex edge edge vertex vertex 3-Feb-14 47

48 Path Connected Any two vertexes can be linked by a path 3-Feb-14 48

49 Rethinking Path Connectivity 0-sphere Let s call this complex 0-connected 1-disc

50 1-Connectivity 1-sphere 2-disc

51 This Complex is not 1- Connected?

52 2-Connectivity 2-sphere 3-disk

53 n-connectivity C is n-connected, if, for m n, every continuous map of the m-sphere f : S m! C can be extended to a continuous map of the (m+1)-disk f : D m+1! C 3-Feb (-1)-connected is non-empty

54 Road Map Simplicial Complexes Standard Constructions Carrier Maps Connectivity Subdivisions Simplicial & Continuous Approximations 54

55 Subdivisions 3-Feb-14 55

56 B is a subdivision of A if For each simplex of B there is a simplex of A such that µ. For each simplex of A, is the union of a finite set of geometric simplexes of B. 3-Feb-14 56

57 Stellar Subdivision Any subdivision is the composition of stellar subdivisions ¾ Stel ¾ 3-Feb-14 57

58 Barycentric Subdivision ¾ Bary ¾ 3-Feb-14 58

59 Barycentric Subdivision ¾ Each vertex of Bary ¾ is a face of ¾ Simplex = faces ordered by inclusion 3-Feb-14 59

60 Barycentric Coordinates x = t 0 v 0 + t 1 v 1 + t 2 v 2 v 0 0 t 0,t 1,t 2 1 t i = 1 x v 2 Every point of C has a unique representation using barycentric coordinates 3-Feb v 1

61 Standard Chromatic Subdivision Ch ¾ Chromatic form of Barycentric

62 Road Map Simplicial Complexes Standard Constructions Carrier Maps Connectivity Subdivisions Simplicial & Continuous Approximations 62

63 From Simplicial to Continuous simplicial Á :A! B f(x) = X i t i já(s i )j continuous f :jaj! jbj extend over barycentric coordinates (piece-wise linear map) 3-Feb-14 63

64 Maps simplicial Á :A! B continuous f :jaj! jbj Simplicial Approximation Theorem 3-Feb-14 64

65 Simplicial Approximation A B simplicial Á : A! B continuous f : jaj! jbj 3-Feb-14 65

66 Simplicial Approximation ~v Á(~v) Á A B 3-Feb-14 66

67 Simplicial Approximation f ~v f(~v) Á(~v) Á A B 3-Feb-14 67

68 Simplicial Approximation ~v f; Á f(~v) Á(~v) A St(Á(~v)) B 3-Feb-14 68

69 Simplicial Approximation ~v f; Á f(~v) A St(Á(~v)) B 3-Feb-14 69

70 Simplicial Approximation ~v f; Á f(~v) A St(~v) St(Á(~v)) 3-Feb B

71 Simplicial Approximation f; Á f(~v) A St(~v) St(Á(~v)) 3-Feb B

72 Simplicial Approximation St(~v) f f(st(~v)) St(Á(~v)) A B 3-Feb-14 72

73 Simplicial Approximation Á is a simplicial approximation of f if for every v in A f(st(~v)) f(st(~v)) µ St(Á(~v)) B 3-Feb-14 73

74 Simplicial Approximation Theorem Given a continuous map f : jaj! jbj there is an N such that f has a simplicial approximation Á : Bary N A! B Actually Holds for most other subdivisions. 3-Feb-14 74

75 This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 Unported License. 75