Isabel K. Darcy. hup://

Transcription

1 Lecture 5: Triangula0ons & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH: 7450 (22M:305) Topics in Topology: Scien0fic and Engineering Applica0ons of Algebraic Topology Target Audience: Anyone interested in topological data analysis including graduate students, faculty, industrial researchers in bioinforma0cs, biology, business, computer science, cosmology, engineering, imaging, mathema0cs, neurology, physics, sta0s0cs, etc. Isabel K. Darcy Mathema0cs Department/Applied Mathema0cal & Computa0onal Sciences niversity of Iowa hp://

2 Building blocks for a simplicial complex 0- simplex vertex v 1- simplex edge {v 1, v 2 } v 1 v 2 e Note that the boundary of this edge is v 2 + v 1 2- simplex triangle {v 1, v 2, v 3 } e 1 v 2 e 2 v 1 v e 3 3 Note that the boundary of this triangle is the cycle e 1 + e 2 + e 3 {v 1, v 2 } + {v 2, v 3 } + {v 1, v 3 }

3 Building blocks for a simplicial complex 3- simplex {v 1, v 2, v 3, v 4 } tetrahedron v 2 v 1 v 3 boundary of {v 1, v 2, v 3, v 4 } {v 1, v 2, v 3 } + {v 1, v 2, v 4 } + {v 1, v 3, v 4 } + {v 2, v 3, v 4 } n- simplex {v 1, v 2,, v n+1 } v 4

4 Building blocks for a simplicial complex 3- simplex {v 1, v 2, v 3, v 4 } tetrahedron v 2 Fill in v 2 v 4 v 4 v 1 v 3 boundary of {v 1, v 2, v 3, v 4 } {v 1, v 2, v 3 } + {v 1, v 2, v 4 } + {v 1, v 3, v 4 } + {v 2, v 3, v 4 } n- simplex {v 1, v 2,, v n+1 } v 1 v 3

5 Crea0ng a simplicial complex 0.) Start by adding 0- dimensional ver0ces (0- simplices)

6 Crea0ng a simplicial complex 1.) Next add 1- dimensional edges (1- simplices). Note: These edges must connect two ver0ces. I.e., the boundary of an edge is two ver0ces

7 Crea0ng a simplicial complex 1.) Next add 1- dimensional edges (1- simplices). Note: These edges must connect two ver0ces. I.e., the boundary of an edge is two ver0ces

8 Crea0ng a simplicial complex 1.) Next add 1- dimensional edges (1- simplices). Note: These edges must connect two ver0ces. I.e., the boundary of an edge is two ver0ces

9 Crea0ng a simplicial complex 2.) Add 2- dimensional triangles (2- simplices). Boundary of a triangle a cycle consis0ng of 3 edges.

10 Crea0ng a simplicial complex 2.) Add 2- dimensional triangles (2- simplices). Boundary of a triangle a cycle consis0ng of 3 edges.

11 Crea0ng a simplicial complex 3.) Add 3- dimensional tetrahedrons (3- simplices). Boundary of a 3- simplex a cycle consis0ng of its four 2- dimensional faces.

12 Crea0ng a simplicial complex 3.) Add 3- dimensional tetrahedrons (3- simplices). Boundary of a 3- simplex a cycle consis0ng of its four 2- dimensional faces.

13 4.) Add 4- dimensional 4- simplices, {v 1, v 2,, v 5 }. Boundary of a 4- simplex a cycle consis0ng of 3- simplices. {v 2, v 3, v 4, v 5 } + {v 1, v 3, v 4, v 5 } + {v 1, v 2, v 4, v 5 } + {v 1, v 2, v 3, v 5 } + {v 1, v 2, v 3, v 4 }

14 Crea0ng a simplicial complex n.) Add n- dimensional n- simplices, {v 1, v 2,, v n+1 }. Boundary of a n- simplex a cycle consis0ng of (n- 1)- simplices.

15 Example: Triangula0ng the circle. circle { x in R 2 : x 1 }

16 Example: Triangula0ng the circle. circle { x in R 2 : x 1 }

17 Example: Triangula0ng the circle. circle { x in R 2 : x 1 }

18 Example: Triangula0ng the circle. circle { x in R 2 : x 1 }

19 Example: Triangula0ng the circle. circle { x in R 2 : x 1 }

20 Example: Triangula0ng the disk. disk { x in R 2 : x 1 }

21 Example: Triangula0ng the disk. disk { x in R 2 : x 1 }

22 Example: Triangula0ng the disk. disk { x in R 2 : x 1 }

23 Example: Triangula0ng the disk. disk { x in R 2 : x 1 }

24 Example: Triangula0ng the disk. disk { x in R 2 : x 1 }

25 Example: Triangula0ng the sphere. sphere { x in R 3 : x 1 }

26 Example: Triangula0ng the sphere. sphere { x in R 3 : x 1 }

27 Example: Triangula0ng the sphere. sphere { x in R 3 : x 1 }

28 Example: Triangula0ng the sphere. sphere { x in R 3 : x 1 }

29 Example: Triangula0ng the circle. disk { x in R 2 : x 1 }

30 Example: Triangula0ng the circle. disk { x in R 2 : x 1 }

31 Example: Triangula0ng the circle. disk { x in R 2 : x 1 } Fist image from hp://openclipart.org/detail/1000/a- raised- fist- by- licarn

32 Example: Triangula0ng the sphere. sphere { x in R 3 : x 1 }

33 Example: Triangula0ng the sphere. sphere { x in R 3 : x 1 }

34 Crea0ng a cell complex Building block: n- cells { x in R n : x 1 } Examples: 0- cell { x in R 0 : x < 1 } 1- cell open interval { x in R : x < 1 } ( ) 2- cell open disk { x in R 2 : x < 1 } 3- cell open ball { x in R 3 : x < 1 }

35 Building blocks for a simplicial complex 0- simplex vertex v 1- simplex edge {v 1, v 2 } v 1 v 2 e Note that the boundary of this edge is v 2 + v 1 2- simplex triangle {v 1, v 2, v 3 } e 1 v 2 e 2 v 1 v e 3 3 Note that the boundary of this triangle is the cycle e 1 + e 2 + e 3 {v 1, v 2 } + {v 2, v 3 } + {v 1, v 3 }

36 Building blocks for a simplicial complex 3- simplex {v 1, v 2, v 3, v 4 } tetrahedron v 2 Fill in v 2 v 4 v 4 v 1 v 3 boundary of {v 1, v 2, v 3, v 4 } {v 1, v 2, v 3 } + {v 1, v 2, v 4 } + {v 1, v 3, v 4 } + {v 2, v 3, v 4 } n- simplex {v 1, v 2,, v n+1 } v 1 v 3

37 Crea0ng a cell complex Building block: n- cells { x in R n : x 1 } Examples: 0- cell { x in R 0 : x < 1 } 1- cell open interval { x in R : x < 1 } ( ) 2- cell open disk { x in R 2 : x < 1 } 3- cell open ball { x in R 3 : x < 1 }

38 Crea0ng a cell complex Building block: n- cells { x in R n : x 1 } Examples: 0- cell { x in R 0 : x < 1 } 1- cell open interval { x in R : x < 1 } ( ) 2- cell open disk { x in R 2 : x < 1 } 3- cell open ball { x in R 3 : x < 1 }

39 Example: disk { x in R 2 : x 1 } Simplicial complex Cell complex ( )

40 Example: disk { x in R 2 : x 1 } Simplicial complex Cell complex ( )

41 Example: disk { x in R 2 : x 1 } Simplicial complex Cell complex ( ) ( )

42 Example: disk { x in R 2 : x 1 } Simplicial complex Cell complex ( ) [ ]

43 Example: disk { x in R 2 : x 1 } Simplicial complex Cell complex ( ) [ ]

44 Example: disk { x in R 2 : x 1 } Simplicial complex Cell complex ( ) [ ]

45 Example: disk { x in R 2 : x 1 } Simplicial complex Cell complex ( ) [ ]

46 Euler characteris?c (simple form): number of ver0ces number of edges + number of faces Or in short- hand, V - E + F where V set of ver0ces E set of edges F set of 2- dimensional faces & the nota0on X the number of elements in the set X.

47 Example: disk { x in R 2 : x 1 } Simplicial complex 3 ver0ces, 3 edges, 1 triangle Cell complex 1 vertex, 1 edge, 1 disk. ( ) [ ]

48 Euler characteris?c: Given a simplicial complex C, let C n the set of n- dimensional simplices in C, and let C n denote the number of elements in C n. Then C 0 - C 1 + C 2 - C 3 + Σ (- 1) n C n

49 Euler characteris?c: Given a cell complex C, let C n the set of n- dimensional cells in C, and let C n denote the number of elements in C n. Then C 0 - C 1 + C 2 - C 3 + Σ (- 1) n C n

50 Example: sphere { x in R 3 : x 1 } Simplicial complex 4 ver0ces, 6 edges, 4 triangles Cell Complex 1 vertex, 1 disk

51 Example: sphere { x in R 3 : x 1 } Simplicial complex Cell complex

52 Example: sphere { x in R 3 : x 1 } Simplicial complex Cell complex

53 Example: sphere { x in R 3 : x 1 } Simplicial complex Cell complex

54 Example: sphere { x in R 3 : x 1 } Simplicial complex Cell complex Fist image from hp://openclipart.org/detail/1000/a- raised- fist- by- licarn

55 Example: sphere { x in R 3 : x 1 } Cell complex

56 Example: sphere { x in R 3 : x 1 } Cell complex

57 Example: sphere { x in R 3 : x 1 } Simplicial complex Cell complex Cell complex