1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 9, 2013: Create your own homology. Fall 2013 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics Applied Mathematical and Computational Sciences, University of Iowa http://www.math.uiowa.edu/~idarcy/AppliedTopology.html

2. Homology 3 ingredients: 1.) Objects 2.) Grading 3.) Boundary map

4. v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 2-simplex = triangle = {v 1, v 2, v 3 } 1-simplex = edge = {v 1, v 2 } e v1v1 v2v2 0-simplex = vertex = v Building blocks for a simplicial complex

5. v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 2-simplex = triangle 1-simplex = edge e v1v1 v2v2 0-simplex = vertex = v Building blocks for a simplicial complex

6. 2-simplex = triangle = {v 1, v 2, v 3 } 1-simplex = edge = {v 1, v 2 } 0-simplex = vertex = v Building blocks for a simplicial complex

7. Objects = generators Generators = X = { x  |  in A } Let R be a ring (or field) R[X] = { n 1 x 1 + n 2 x 2 + … + n k x k : x i in R } Ex: Z 2 [X] = { n 1 x 1 + n 2 x 2 + … + n k x k : x i in Z 2 }

8. A free vector space over the field F generated by the elements x 1, x 2, …, x k consists of all elements of the form n 1 x 1 + n 2 x 2 + … + n k x k where n i in F. Examples of a field: R = set of real numbers Q = set of rational numbers Z 2 = {0, 1} Addition: (n 1 x 1 + n 2 x 2 + … + n k x k ) + (m 1 x 1 + m 2 x 2 + … + m k x k ) = (n 1 + m 1 ) x 1 + (n 2 + m 2 )x 2 + … + (n k + m k )x k Slide from preparatory lecture 4: Addition (and free vector spaces)

9. A free vector space over the field F generated by the elements x 1, x 2, …, x k consists of all elements of the form n 1 x 1 + n 2 x 2 + … + n k x k where n i in F. Examples of a field: R = set of real numbers: πx + √2 y + 3z is in R[x, y, z] Q = set of rational numbers (i.e. fractions): (½)x + 4y is in Q[x, y] Z 2 = {0, 1}: 0x + 1y + 1w + 0z is in Z 2 [x, y, z, w] Slide from preparatory lecture 4: Addition (and free vector spaces)

10. A free abelian group generated by the elements x 1, x 2, …, x k consists of all elements of the form n 1 x 1 + n 2 x 2 + … + n k x k where n i are integers. Example: Z[ x, x ] 4 ix + 2 I x – 2 i -3 x k + n iii Z = The set of integers = { …, -2, -1, 0, 1, 2, …}

12. Grading Grading: Each object is assigned a unique grade Grading = Partition of R[x] Ex: Grade = dimension v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 Grade 2: 2-simplex = triangle = {v 1, v 2, v 3 } Grade 1: 1-simplex = edge = {v 1, v 2 } e v1v1 v2v2 Grade 0: 0-simplex = vertex = v

13. Grading Grading: Each object is assigned a unique grade Grading = Partition of R[x] Ex: Grade: Cardinality Grade 2: 2-simplex = triangle = {v 1, v 2, v 3 } Grade 3: 3-simplex = tetrahedron = {v 1, v 2, v 3, v 4 } Grade 1: 1-simplex = edge = {v 1, v 2 } Grade 0: 0-simplex = vertex = {v}

14. Grading Grading: Each object is assigned a unique grade. Grading = Partition of R[x]. Let X n = {x 1, …, x k } = generators of grade n. C n = set of n-chains = R[X n ]

16. Boundary Map n : C n  C n-1 such that 2 = 0 v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 e v1v1 v2v2   0 v1v1 v2v2 v2v2 e2e2 e1e1 e3e3 v1v1 v3v3

17. {v 1, v 2 }  {v 1 } + {v 2 }  0 {v 1, v 2, v 3 }  {v 1, v 2 } + {v 1, v 3 } + {v 2, v 3 } Boundary Map n : C n  C n-1 such that 2 = 0

18. C n+1  C n  C n-1 ...  C 2  C 1  C 0  0 H n = Z n /B n = (kernel of )/ (image of ) null space of M n column space of M n+1 Rank H n = Rank Z n – Rank B n = n+1 n n210

19. Your name homology 3 ingredients: 1.) Objects 2.) Grading 3.) Boundary map

20. v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 2-simplex = face = {v 1, v 2, v 3 } Note that the boundary of this face is the cycle e 1 + e 2 + e 3 = {v 1, v 2 } + {v 2, v 3 } + {v 1, v 3 } 1-simplex = edge = {v 1, v 2 } Note that the boundary of this edge is v 2 + v 1 e v1v1 v2v2 0-simplex = vertex = v Unoriented simplicial complex using Z 2 coefficients Grading = dimension

21. v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 2-simplex = oriented face = (v 1, v 2, v 3 ) 1-simplex = oriented edge = (v 1, v 2 ) Note that the boundary of this edge is v 2 – v 1 e v1v1 v2v2 0-simplex = vertex = v Note that the boundary of this face is the cycle e 1 + e 2 + e 3 = (v 1, v 2 ) + (v 2, v 3 ) – (v 1, v 3 ) = (v 1, v 2 ) – (v 1, v 3 ) + (v 2, v 3 ) Oriented simplicial complex Grading = dimension

22. Cell complex Building block: n-cells = { x in R n : || x || ≤ 1 } 2-cell = open disk = { x in R 2 : ||x || < 1 } Examples: 0-cell = { x in R 0 : ||x || < 1 } 1-cell =open interval ={ x in R : ||x || < 1 } ( ) Grading = dimension (n-cells) = { x in R n : || x || = 1 } n

23. Čech homology Given U V   where  V   open for all  in A. Objects = finite intersections = { V   i in A } Grading = n = depth of intersection. ( V   V  Ex: (V     V  V   V   V   V  V  V   V   V     V   V     V   V   U i = 1 n i  in A n+1 j = 1 n i i U i = 1 i ≠ j n U i = 1 n ( ) 0 1 U UUUUU 2

24. Creating the Čech simplicial complex

25. Consider X an arbitrary topological space. Let V = {V i | i = 1, …, n } where V i X, The nerve of V = N(V) where The k -simplices of N(V) = nonempty intersections of k +1 distinct elements of V. For example, Vertices = elements of V Edges = pairs in V which intersect nontrivially. Triangle = triples in V which intersect nontrivially. http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf

26. Nerve Lemma: If V is a finite collection of subsets of X with all non-empty intersections of subcollections of V contractible, then N(V) is homotopic to the union of elements of V. http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf

27. Creating the Vietoris Rips simplicial complex

29. Betti numbers provide a signature of the underlying topology. Singh G et al. J Vis 2008;8:11 ©2008 by Association for Research in Vision and Ophthalmology

30. From: http://www.math.cornell.edu/~mec/Winter2009/Victor/part1.htm

31. From: http://plus.maths.org/content/imaging-maths-inside-klein-bottle From: http://www.math.osu.edu/~fiedorowicz.1/math655/Klein2.html Klein Bottle

32. Betti numbers provide a signature of the underlying topology. Singh G et al. J Vis 2008;8:11 ©2008 by Association for Research in Vision and Ophthalmology

33. http://www.journalofvision.org/ content/8/8/11.full Figure 4 animation

34. Topology and Data. Gunnar Carlsson www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X

35. v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 v6v6 v4v4 v5v5 e4e4 e6e6 e5e5

36. v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 v6v6 v4v4 v5v5 e4e4 e6e6 e5e5

37. v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 v6v6 v4v4 v5v5 e4e4 e6e6 e5e5 Z 1 = kernel of = null space of M 1 = o1o1

38. v2v2 e2e2 e1e1 e3e3 v1v1 v3v3 v6v6 v4v4 v5v5 e4e4 e6e6 e5e5

39. C 2  C 1  C 0 o1o1 o2o2 H 1 = Z 1 /B 1 = (kernel of )/ (image of ) null space of M 1 column space of M 2 Rank H 1 = Rank Z 1 – Rank B 1 = 2 – 1 = 1 o1o1 o2o2 = = v6v6 v4v4 v5v5 e3e3 e5e5 e4e4 v3v3 v1v1 v2v2 e1e1 e3e3 e2e2