Lecture 5: Triangulations & simplicial complexes (and cell complexes).

1. Isabel K. Darcy Mathematics Department/Applied Mathematical & Computational Sciences University of Iowa http://www.math.uiowa.edu/~idarcy/AppliedTopology.html Lecture 5: Triangulations & simplicial complexes (and cell complexes). in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Target Audience: Anyone interested in topological data analysis including graduate students, faculty, industrial researchers in bioinformatics, biology, business, computer science, cosmology, engineering, imaging, mathematics, neurology, physics, statistics, etc.

2. Building blocks for a simplicial complex e 0-simplex = vertex = v 1-simplex = edge = {v1, v2} Note that the boundary of this edge is v2+ v1 2-simplex = triangle = {v1, v2, v3} v2 Note that the boundary of this triangle is the cycle e1 + e2 + e3 = {v1, v2} + {v2, v3} +{v1, v3} e1 e2 v1 v1 v3 v2 e3

3. Building blocks for a simplicial complex 3-simplex = {v1, v2, v3, v4} = tetrahedron boundary of {v1, v2, v3, v4} = {v1, v2, v3} + {v1, v2, v4} + {v1, v3, v4} + {v2, v3, v4} n-simplex = {v1, v2, …, vn+1} v2 v4 v1 v3

4. Building blocks for a simplicial complex 3-simplex = {v1, v2, v3, v4} = tetrahedron boundary of {v1, v2, v3, v4} = {v1, v2, v3} + {v1, v2, v4} + {v1, v3, v4} + {v2, v3, v4} n-simplex = {v1, v2, …, vn+1} v2 v2 Fill in v4 v4 v1 v1 v3 v3

5. Creating a simplicial complex 0.) Start by adding 0-dimensional vertices (0-simplices)

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

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

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

9. Creating a simplicial complex 2.) Add 2-dimensional triangles (2-simplices). Boundary of a triangle = a cycle consisting of 3 edges.

10. Creating a simplicial complex 2.) Add 2-dimensional triangles (2-simplices). Boundary of a triangle = a cycle consisting of 3 edges.

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

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

13. 4.) Add 4-dimensional 4-simplices, {v1, v2, …, v5}. Boundary of a 4-simplex = a cycle consisting of 3-simplices. = {v2, v3, v4, v5} + {v1, v3, v4, v5} + {v1, v2, v4, v5} + {v1, v2, v3, v5} + {v1, v2, v3, v4}

14. Creating a simplicial complex n.) Add n-dimensional n-simplices, {v1, v2, …, vn+1}. Boundary of a n-simplex = a cycle consisting of (n-1)-simplices.

15. Example: Triangulating the circle. circle = { x in R2 : ||x || = 1 }

16. Example: Triangulating the circle. circle = { x in R2 : ||x || = 1 }

17. Example: Triangulating the circle. circle = { x in R2 : ||x || = 1 }

18. Example: Triangulating the circle. circle = { x in R2 : ||x || = 1 }

19. Example: Triangulating the circle. circle = { x in R2 : ||x || = 1 }

20. Example: Triangulating the disk. disk = { x in R2 : ||x || ≤ 1 }

21. Example: Triangulating the disk. disk = { x in R2 : ||x || ≤ 1 }

22. Example: Triangulating the disk. disk = { x in R2 : ||x || ≤ 1 }

23. Example: Triangulating the disk. disk = { x in R2 : ||x || ≤ 1 }

24. Example: Triangulating the disk. disk = { x in R2 : ||x || ≤ 1 } =

25. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 }

26. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 }

27. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 }

28. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 }

29. Example: Triangulating the circle. disk = { x in R2 : ||x || ≤ 1 } =

30. Example: Triangulating the circle. disk = { x in R2 : ||x || ≤ 1 }

31. Example: Triangulating the circle. disk = { x in R2 : ||x || ≤ 1 } Fist image from http://openclipart.org/detail/1000/a-raised-fist-by-liftarn

32. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 }

33. Example: Triangulating the sphere. sphere = { x in R3 : ||x || = 1 } =

34. Creating a cell complex Building block: n-cells = { x in Rn : || x || ≤ 1 } Examples: 0-cell = { x in R0 : ||x || < 1 } 1-cell =open interval ={ x in R : ||x || < 1 } ( ) 2-cell = open disk = { x in R2 : ||x || < 1 } 3-cell = open ball = { x in R3 : ||x || < 1 }

35. Building blocks for a simplicial complex e 0-simplex = vertex = v 1-simplex = edge = {v1, v2} Note that the boundary of this edge is v2+ v1 2-simplex = triangle = {v1, v2, v3} v2 Note that the boundary of this triangle is the cycle e1 + e2 + e3 = {v1, v2} + {v2, v3} +{v1, v3} e1 e2 v1 v1 v3 v2 e3

36. Building blocks for a simplicial complex 3-simplex = {v1, v2, v3, v4} = tetrahedron boundary of {v1, v2, v3, v4} = {v1, v2, v3} + {v1, v2, v4} + {v1, v3, v4} + {v2, v3, v4} n-simplex = {v1, v2, …, vn+1} v2 v2 Fill in v4 v4 v1 v1 v3 v3

37. Creating a cell complex Building block: n-cells = { x in Rn : || x || ≤ 1 } Examples: 0-cell = { x in R0 : ||x || < 1 } 1-cell =open interval ={ x in R : ||x || < 1 } ( ) 2-cell = open disk = { x in R2 : ||x || < 1 } 3-cell = open ball = { x in R3 : ||x || < 1 }

38. Creating a cell complex Building block: n-cells = { x in Rn : || x || ≤ 1 } Examples: 0-cell = { x in R0 : ||x || < 1 } 1-cell =open interval ={ x in R : ||x || < 1 } ( ) 2-cell = open disk = { x in R2 : ||x || < 1 } 3-cell = open ball = { x in R3 : ||x || < 1 }

39. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) Cell complex = U U

40. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) Cell complex = U U

41. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) ( ) Cell complex = U U U

42. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) [] Cell complex = U U U

43. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) [] Cell complex = U U = U

44. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) [] Cell complex = U U = U U

45. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex = ( ) [] Cell complex = U U = U = U

46. Euler characteristic (simple form): = number of vertices – number of edges + number of faces Or in short-hand, = |V| - |E| + |F| where V = set of vertices E = set of edges F = set of 2-dimensional faces & the notation |X| = the number of elements in the set X.

47. Example: disk = { x in R2 : ||x || ≤ 1 } Simplicial complex 3 vertices, 3 edges, 1 triangle = Cell complex 1 vertex, 1 edge, 1 disk. ( ) [] = U U = U = U

48. Euler characteristic: Given a simplicial complex C, let Cn = the set of n-dimensional simplices in C, and let |Cn| denote the number of elements in Cn. Then = |C0| - |C1| + |C2| - |C3| + … = Σ (-1)n |Cn|

49. Euler characteristic: Given a cell complex C, let Cn = the set of n-dimensional cells in C, and let |Cn| denote the number of elements in Cn. Then = |C0| - |C1| + |C2| - |C3| + … = Σ (-1)n |Cn|

50. Example: sphere = { x in R3 : ||x || = 1 } = Simplicial complex 4 vertices, 6 edges, 4 triangles Cell Complex 1 vertex, 1 disk = U