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Introduction to Computational Topology Ahmed Abdelkader Guest Lecture CMSC 754 Spring 2020 May 7th, 2020 Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 1 / 41 Introduction Pages 13 Early Topological

  1. Introduction Pages 1–3 Convex Polytopes 4 − 6 + 4 8 − 12 + 6 6 − 12 + 8 20 − 30 + 12 12 − 30 + 20 Euler’s Polyhedron Formula Alternating sum of the number of vertices (V), edges (E), and facets (F) χ = V − E + F As spheres can be continuously deformed into convex polytopes, they also have an Euler characteristic of 2. Unlike the genus, this is easily computed by simple counting or algebra. Figures from Wikipedia [10, 11, 12, 13, 14] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 6 / 41

  2. Introduction Pages 1–3 Convex Polytopes 4 − 6 + 4 8 − 12 + 6 6 − 12 + 8 20 − 30 + 12 12 − 30 + 20 Euler’s Polyhedron Formula Alternating sum of the number of vertices (V), edges (E), and facets (F) χ = V − E + F As spheres can be continuously deformed into convex polytopes, they also have an Euler characteristic of 2. Unlike the genus, this is easily computed by simple counting or algebra. Figures from Wikipedia [10, 11, 12, 13, 14] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 6 / 41

  3. Preliminaries Page 4 What about non-convex surfaces? Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 7 / 41

  4. Preliminaries Page 4 Wireframes Rendering all triangles Wireframe, edges only Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 8 / 41

  5. Preliminaries Page 4 Simplicial Complexes A 3-simplex Four 2-simplices Six 1-simplices Four 0-simplices Definitions A p-simplex is the convex hull of ( p + 1) affinely-independent points. We write this as σ = [ v 0 , . . . , v p ] = conv { v 0 , . . . , v p } and say dim σ = p . A simplicial complex K is a set of simplices closed under intersection, and its dimension dim K is the maximum dimension of its simplices. If σ 1 , σ 2 ∈ K , then σ 1 ∩ σ 2 ∈ K . The ( − 1)-simplex ∅ is always in K . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 9 / 41

  6. Preliminaries Page 4 Simplicial Complexes A 3-simplex Four 2-simplices Six 1-simplices Four 0-simplices Definitions A p-simplex is the convex hull of ( p + 1) affinely-independent points. We write this as σ = [ v 0 , . . . , v p ] = conv { v 0 , . . . , v p } and say dim σ = p . A simplicial complex K is a set of simplices closed under intersection, and its dimension dim K is the maximum dimension of its simplices. If σ 1 , σ 2 ∈ K , then σ 1 ∩ σ 2 ∈ K . The ( − 1)-simplex ∅ is always in K . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 9 / 41

  7. Preliminaries Page 4 Simplicial Complexes A 3-simplex Four 2-simplices Six 1-simplices Four 0-simplices Definitions A p-simplex is the convex hull of ( p + 1) affinely-independent points. We write this as σ = [ v 0 , . . . , v p ] = conv { v 0 , . . . , v p } and say dim σ = p . A simplicial complex K is a set of simplices closed under intersection, and its dimension dim K is the maximum dimension of its simplices. If σ 1 , σ 2 ∈ K , then σ 1 ∩ σ 2 ∈ K . The ( − 1)-simplex ∅ is always in K . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 9 / 41

  8. Preliminaries Page 4 Simplicial Complexes A 3-simplex Four 2-simplices Six 1-simplices Four 0-simplices Definitions A p-simplex is the convex hull of ( p + 1) affinely-independent points. We write this as σ = [ v 0 , . . . , v p ] = conv { v 0 , . . . , v p } and say dim σ = p . A simplicial complex K is a set of simplices closed under intersection, and its dimension dim K is the maximum dimension of its simplices. If σ 1 , σ 2 ∈ K , then σ 1 ∩ σ 2 ∈ K . The ( − 1)-simplex ∅ is always in K . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 9 / 41

  9. Preliminaries Page 4 Simplicial Complexes A 3-simplex Four 2-simplices Six 1-simplices Four 0-simplices Definitions A face τ is a k -simplex connecting ( k + 1) of the vertices of σ . We write this as τ � σ , and say that σ is a coface of τ . A (co)face τ of a simplex σ is proper if dim τ � = dim σ . The boundary ∂σ is the collection of proper faces of σ The interior of σ is defined as | σ | = σ − ∂σ . The underlying space of a complex K is defined as | K | = ∪ σ ∈ K | σ | . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 9 / 41

  10. Preliminaries Page 4 Simplicial Complexes A 3-simplex Four 2-simplices Six 1-simplices Four 0-simplices Definitions A face τ is a k -simplex connecting ( k + 1) of the vertices of σ . We write this as τ � σ , and say that σ is a coface of τ . A (co)face τ of a simplex σ is proper if dim τ � = dim σ . The boundary ∂σ is the collection of proper faces of σ The interior of σ is defined as | σ | = σ − ∂σ . The underlying space of a complex K is defined as | K | = ∪ σ ∈ K | σ | . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 9 / 41

  11. Preliminaries Page 4 Simplicial Complexes A 3-simplex Four 2-simplices Six 1-simplices Four 0-simplices Definitions A face τ is a k -simplex connecting ( k + 1) of the vertices of σ . We write this as τ � σ , and say that σ is a coface of τ . A (co)face τ of a simplex σ is proper if dim τ � = dim σ . The boundary ∂σ is the collection of proper faces of σ The interior of σ is defined as | σ | = σ − ∂σ . The underlying space of a complex K is defined as | K | = ∪ σ ∈ K | σ | . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 9 / 41

  12. Preliminaries Page 4 Simplicial Complexes A 3-simplex Four 2-simplices Six 1-simplices Four 0-simplices Definitions A face τ is a k -simplex connecting ( k + 1) of the vertices of σ . We write this as τ � σ , and say that σ is a coface of τ . A (co)face τ of a simplex σ is proper if dim τ � = dim σ . The boundary ∂σ is the collection of proper faces of σ The interior of σ is defined as | σ | = σ − ∂σ . The underlying space of a complex K is defined as | K | = ∪ σ ∈ K | σ | . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 9 / 41

  13. Preliminaries Page 4 Simplicial Complexes A 3-simplex Four 2-simplices Six 1-simplices Four 0-simplices Definitions A face τ is a k -simplex connecting ( k + 1) of the vertices of σ . We write this as τ � σ , and say that σ is a coface of τ . A (co)face τ of a simplex σ is proper if dim τ � = dim σ . The boundary ∂σ is the collection of proper faces of σ The interior of σ is defined as | σ | = σ − ∂σ . The underlying space of a complex K is defined as | K | = ∪ σ ∈ K | σ | . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 9 / 41

  14. Simplicial Approximations Pages 5–7 How to represent a mapping between two surfaces? Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 10 / 41

  15. Simplicial Approximations Pages 5–7 Continuous Deformations A continuous deformation of a cow model into a ball Figure from Wikipedia [15] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 11 / 41

  16. Simplicial Approximations Pages 5–7 Continuous Maps Continuity at x = 2 by ( ε, δ ) Continuity at x ∈ X using neighborhoods Definition of Continuity Small changes in the input yield small changes in the output. Calculus formalizes this notion using the ( ε, δ )-definition of the limit. For general topologies, we use neighborhoods instead of ( ε, δ ) intervals. Figures from Wikipedia [16, 17] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 12 / 41

  17. Simplicial Approximations Pages 5–7 Continuous Maps Continuity at x = 2 by ( ε, δ ) Continuity at x ∈ X using neighborhoods Definition of Continuity Small changes in the input yield small changes in the output. Calculus formalizes this notion using the ( ε, δ )-definition of the limit. For general topologies, we use neighborhoods instead of ( ε, δ ) intervals. Figures from Wikipedia [16, 17] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 12 / 41

  18. Simplicial Approximations Pages 5–7 Continuous Maps Continuity at x = 2 by ( ε, δ ) Continuity at x ∈ X using neighborhoods Definition of Continuity Small changes in the input yield small changes in the output. Calculus formalizes this notion using the ( ε, δ )-definition of the limit. For general topologies, we use neighborhoods instead of ( ε, δ ) intervals. Figures from Wikipedia [16, 17] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 12 / 41

  19. Simplicial Approximations Pages 5–7 Homeomorphisms f f − 1 Definition Two topological spaces X and Y are said to be homeomorphic whenever there exists a continuous map f : X → Y with a continuous inverse f − 1 : Y → X . Such a function f is called a homeomorphism . Figure from Wikipedia [15] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 13 / 41

  20. Simplicial Approximations Pages 5–7 But, we will be using the triangulations rather than the surfaces ... Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 14 / 41

  21. Simplicial Approximations Pages 5–7 Triangulations Definition A triangulation of a topological space X is a simplicial complex ˆ X such that X and | ˆ X | are homeomorphic. A topological space is triangulable if it admits a triangulation. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 15 / 41

  22. Simplicial Approximations Pages 5–7 Triangulations Definition A triangulation of a topological space X is a simplicial complex ˆ X such that X and | ˆ X | are homeomorphic. A topological space is triangulable if it admits a triangulation. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 15 / 41

  23. Simplicial Approximations Pages 5–7 Continuous Maps between Simplicial Complexes Simplicial Neighborhoods Fix a simplicial complex K . The star of σ is the collection its cofaces: St K ( σ ) = { τ ∈ K | σ � τ } . The star neighborhood of σ is the union of the interior of its cofaces: N K ( σ ) = ∪ τ ∈ St K ( σ ) | τ | . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 16 / 41

  24. Simplicial Approximations Pages 5–7 Continuous Maps between Simplicial Complexes Simplicial Neighborhoods Fix a simplicial complex K . The star of σ is the collection its cofaces: St K ( σ ) = { τ ∈ K | σ � τ } . The star neighborhood of σ is the union of the interior of its cofaces: N K ( σ ) = ∪ τ ∈ St K ( σ ) | τ | . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 16 / 41

  25. Simplicial Approximations Pages 5–7 Continuous Maps between Simplicial Complexes Simplicial Neighborhoods Fix a simplicial complex K . The star of σ is the collection its cofaces: St K ( σ ) = { τ ∈ K | σ � τ } . The star neighborhood of σ is the union of the interior of its cofaces: N K ( σ ) = ∪ τ ∈ St K ( σ ) | τ | . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 16 / 41

  26. Simplicial Approximations Pages 5–7 Continuous Maps between Simplicial Complexes X �→ ˆ ˆ Y The Star Condition Fix two simplicial complexes ˆ X and ˆ Y and a map ˆ f : | ˆ X | → | ˆ Y | . We say that ˆ f satisfies the star condition if for all vertices v ∈ ˆ X ˆ for some vertex u = φ ( v ) ∈ ˆ � � f N ˆ X ( v ) ⊆ N ˆ Y ( u ) Y . The map φ : Vert ˆ X → Vert ˆ Y extends to a simplicial map that maps every simplex σ ∈ ˆ X to some simplex τ ∈ ˆ Y . The simplicial map induces a simplicial approximation : a piecewise-linear map ˆ f ∆ : ˆ X → ˆ Y that approximates the original function f . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 16 / 41

  27. Simplicial Approximations Pages 5–7 Continuous Maps between Simplicial Complexes X �→ ˆ ˆ Y The Star Condition Fix two simplicial complexes ˆ X and ˆ Y and a map ˆ f : | ˆ X | → | ˆ Y | . We say that ˆ f satisfies the star condition if for all vertices v ∈ ˆ X ˆ for some vertex u = φ ( v ) ∈ ˆ � � f N ˆ X ( v ) ⊆ N ˆ Y ( u ) Y . The map φ : Vert ˆ X → Vert ˆ Y extends to a simplicial map that maps every simplex σ ∈ ˆ X to some simplex τ ∈ ˆ Y . The simplicial map induces a simplicial approximation : a piecewise-linear map ˆ f ∆ : ˆ X → ˆ Y that approximates the original function f . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 16 / 41

  28. Simplicial Approximations Pages 5–7 Continuous Maps between Simplicial Complexes X �→ ˆ ˆ Y The Star Condition Fix two simplicial complexes ˆ X and ˆ Y and a map ˆ f : | ˆ X | → | ˆ Y | . We say that ˆ f satisfies the star condition if for all vertices v ∈ ˆ X ˆ for some vertex u = φ ( v ) ∈ ˆ � � f N ˆ X ( v ) ⊆ N ˆ Y ( u ) Y . The map φ : Vert ˆ X → Vert ˆ Y extends to a simplicial map that maps every simplex σ ∈ ˆ X to some simplex τ ∈ ˆ Y . The simplicial map induces a simplicial approximation : a piecewise-linear map ˆ f ∆ : ˆ X → ˆ Y that approximates the original function f . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 16 / 41

  29. Simplicial Approximations Pages 5–7 Continuous Maps between Simplicial Complexes X �→ ˆ ˆ Y The Star Condition Fix two simplicial complexes ˆ X and ˆ Y and a map ˆ f : | ˆ X | → | ˆ Y | . We say that ˆ f satisfies the star condition if for all vertices v ∈ ˆ X ˆ for some vertex u = φ ( v ) ∈ ˆ � � f N ˆ X ( v ) ⊆ N ˆ Y ( u ) Y . The map φ : Vert ˆ X → Vert ˆ Y extends to a simplicial map that maps every simplex σ ∈ ˆ X to some simplex τ ∈ ˆ Y . The simplicial map induces a simplicial approximation : a piecewise-linear map ˆ f ∆ : ˆ X → ˆ Y that approximates the original function f . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 16 / 41

  30. Simplicial Approximations Pages 5–7 What if ˆ f : | ˆ X | → | ˆ Y | fails the star condition? Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 17 / 41

  31. Simplicial Approximations Pages 5–7 Simplicial Approximation Theorem Barycentric Subdivisions If there exists a vertex v ∈ ˆ X such that ˆ � � f N ˆ X ( v ) is not contained in Y ( u ) for any vertex u ∈ ˆ N ˆ Y , then N ˆ X ( v ) is too large! Solution: refine ˆ X without changing ˆ f : ˆ X → ˆ Y . � p 1 The barycenter of σ = [ v 0 , . . . , v p ] is defined as i =0 v i . p +1 Repeated subdivisions eventually achieve the star condition. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 18 / 41

  32. Simplicial Approximations Pages 5–7 Simplicial Approximation Theorem Barycentric Subdivisions If there exists a vertex v ∈ ˆ X such that ˆ � � f N ˆ X ( v ) is not contained in Y ( u ) for any vertex u ∈ ˆ N ˆ Y , then N ˆ X ( v ) is too large! Solution: refine ˆ X without changing ˆ f : ˆ X → ˆ Y . � p 1 The barycenter of σ = [ v 0 , . . . , v p ] is defined as i =0 v i . p +1 Repeated subdivisions eventually achieve the star condition. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 18 / 41

  33. Simplicial Approximations Pages 5–7 Simplicial Approximation Theorem Barycentric Subdivisions If there exists a vertex v ∈ ˆ X such that ˆ � � f N ˆ X ( v ) is not contained in Y ( u ) for any vertex u ∈ ˆ N ˆ Y , then N ˆ X ( v ) is too large! Solution: refine ˆ X without changing ˆ f : ˆ X → ˆ Y . � p 1 The barycenter of σ = [ v 0 , . . . , v p ] is defined as i =0 v i . p +1 Repeated subdivisions eventually achieve the star condition. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 18 / 41

  34. Simplicial Approximations Pages 5–7 Simplicial Approximation Theorem Barycentric Subdivisions If there exists a vertex v ∈ ˆ X such that ˆ � � f N ˆ X ( v ) is not contained in Y ( u ) for any vertex u ∈ ˆ N ˆ Y , then N ˆ X ( v ) is too large! Solution: refine ˆ X without changing ˆ f : ˆ X → ˆ Y . � p 1 The barycenter of σ = [ v 0 , . . . , v p ] is defined as i =0 v i . p +1 Repeated subdivisions eventually achieve the star condition. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 18 / 41

  35. Simplicial Approximations Pages 5–7 Simplicial Approximation Theorem Barycentric Subdivisions If there exists a vertex v ∈ ˆ X such that ˆ � � f N ˆ X ( v ) is not contained in Y ( u ) for any vertex u ∈ ˆ N ˆ Y , then N ˆ X ( v ) is too large! Solution: refine ˆ X without changing ˆ f : ˆ X → ˆ Y . � p 1 The barycenter of σ = [ v 0 , . . . , v p ] is defined as i =0 v i . p +1 Repeated subdivisions eventually achieve the star condition. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 18 / 41

  36. Chains Pages 8–9 From Convex Polyhedra to Simplicial Complexes Simplicial Counting Recall the alternating sum used to compute the Euler characteristic χ . We would like to derive a similar computation on a simplicial complex K . But, a single simplex can be shared among multiple cofaces. How do we keep track of the correct count? Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 19 / 41

  37. Chains Pages 8–9 From Convex Polyhedra to Simplicial Complexes Simplicial Counting Recall the alternating sum used to compute the Euler characteristic χ . We would like to derive a similar computation on a simplicial complex K . But, a single simplex can be shared among multiple cofaces. How do we keep track of the correct count? Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 19 / 41

  38. Chains Pages 8–9 From Convex Polyhedra to Simplicial Complexes Simplicial Counting Recall the alternating sum used to compute the Euler characteristic χ . We would like to derive a similar computation on a simplicial complex K . But, a single simplex can be shared among multiple cofaces. How do we keep track of the correct count? Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 19 / 41

  39. Chains Pages 8–9 From Convex Polyhedra to Simplicial Complexes Simplicial Counting Recall the alternating sum used to compute the Euler characteristic χ . We would like to derive a similar computation on a simplicial complex K . But, a single simplex can be shared among multiple cofaces. How do we keep track of the correct count? Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 19 / 41

  40. Chains Pages 8–9 From Convex Polyhedra to Simplicial Complexes Simplicial Counting Recall the alternating sum used to compute the Euler characteristic χ . We would like to derive a similar computation on a simplicial complex K . But, a single simplex can be shared among multiple cofaces. How do we keep track of the correct count? Algebra! Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 19 / 41

  41. Chains Pages 8–9 Chains Counting Modulo 2 Define a p -chain as a subset of the p -simplices in the complex K . We write a p -chain as a formal sum c = � i a i σ i , where σ i ranges over the p -simplices and a i is a coefficient. We will work with coefficients in F 2 = { 0 , 1 } with addition modulo 2 . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 20 / 41

  42. Chains Pages 8–9 Chains Counting Modulo 2 Define a p -chain as a subset of the p -simplices in the complex K . We write a p -chain as a formal sum c = � i a i σ i , where σ i ranges over the p -simplices and a i is a coefficient. We will work with coefficients in F 2 = { 0 , 1 } with addition modulo 2 . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 20 / 41

  43. Chains Pages 8–9 Chains Counting Modulo 2 Define a p -chain as a subset of the p -simplices in the complex K . We write a p -chain as a formal sum c = � i a i σ i , where σ i ranges over the p -simplices and a i is a coefficient. We will work with coefficients in F 2 = { 0 , 1 } with addition modulo 2 . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 20 / 41

  44. Chains Pages 8–9 Chains Counting Modulo 2 Two p -chains can be added to obtain a new p -chain. Letting c 1 = � i a i σ i and c 2 = � i b i σ i . Then, c 1 + c 2 = � i ( a i + b i ) σ i . As a i + b i ∈ F 2 for all i , we get that c 1 + c 2 is a chain. Regarding p -chains as sets, we can interpret that c 1 + c 2 with modulo 2 coefficients is the symmetric difference between the two sets. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 20 / 41

  45. Chains Pages 8–9 Chains Counting Modulo 2 Two p -chains can be added to obtain a new p -chain. Letting c 1 = � i a i σ i and c 2 = � i b i σ i . Then, c 1 + c 2 = � i ( a i + b i ) σ i . As a i + b i ∈ F 2 for all i , we get that c 1 + c 2 is a chain. Regarding p -chains as sets, we can interpret that c 1 + c 2 with modulo 2 coefficients is the symmetric difference between the two sets. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 20 / 41

  46. Chains Pages 8–9 Chains Counting Modulo 2 Two p -chains can be added to obtain a new p -chain. Letting c 1 = � i a i σ i and c 2 = � i b i σ i . Then, c 1 + c 2 = � i ( a i + b i ) σ i . As a i + b i ∈ F 2 for all i , we get that c 1 + c 2 is a chain. Regarding p -chains as sets, we can interpret that c 1 + c 2 with modulo 2 coefficients is the symmetric difference between the two sets. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 20 / 41

  47. Chains Pages 8–9 Chains Counting Modulo 2 Two p -chains can be added to obtain a new p -chain. Letting c 1 = � i a i σ i and c 2 = � i b i σ i . Then, c 1 + c 2 = � i ( a i + b i ) σ i . As a i + b i ∈ F 2 for all i , we get that c 1 + c 2 is a chain. Regarding p -chains as sets, we can interpret that c 1 + c 2 with modulo 2 coefficients is the symmetric difference between the two sets. Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 20 / 41

  48. Chains Pages 8–9 Chain Groups Algebra I A group ( A , • ) is a set A together with a binary operation satisfying: Closure: for all α, β ∈ A , we have that α • β ∈ A . Associativity: so that for all α, β, γ ∈ A we have α • ( β • γ ) = ( α • β ) • γ . A has an identity element ω such that α + ω = α for all α ∈ A . If, in addition, • is commutative , we have that α • β = β • α for all α, β ∈ A , and we say the group ( A , • ) is abelian . Chains as Groups We can now recognize p -chains ( C p , +) as abelian groups. Chains as Vector Spaces If the complex K has n p p -simplices, then C p is (isomorphic to) the set of binary vectors of length n p , i.e., { 0 , 1 } n p , with the exclusive-or operation ⊕ . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 21 / 41

  49. Chains Pages 8–9 Chain Groups Algebra I A group ( A , • ) is a set A together with a binary operation satisfying: Closure: for all α, β ∈ A , we have that α • β ∈ A . Associativity: so that for all α, β, γ ∈ A we have α • ( β • γ ) = ( α • β ) • γ . A has an identity element ω such that α + ω = α for all α ∈ A . If, in addition, • is commutative , we have that α • β = β • α for all α, β ∈ A , and we say the group ( A , • ) is abelian . Chains as Groups We can now recognize p -chains ( C p , +) as abelian groups. Chains as Vector Spaces If the complex K has n p p -simplices, then C p is (isomorphic to) the set of binary vectors of length n p , i.e., { 0 , 1 } n p , with the exclusive-or operation ⊕ . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 21 / 41

  50. Chains Pages 8–9 Chain Groups Algebra I A group ( A , • ) is a set A together with a binary operation satisfying: Closure: for all α, β ∈ A , we have that α • β ∈ A . Associativity: so that for all α, β, γ ∈ A we have α • ( β • γ ) = ( α • β ) • γ . A has an identity element ω such that α + ω = α for all α ∈ A . If, in addition, • is commutative , we have that α • β = β • α for all α, β ∈ A , and we say the group ( A , • ) is abelian . Chains as Groups We can now recognize p -chains ( C p , +) as abelian groups. Chains as Vector Spaces If the complex K has n p p -simplices, then C p is (isomorphic to) the set of binary vectors of length n p , i.e., { 0 , 1 } n p , with the exclusive-or operation ⊕ . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 21 / 41

  51. Chains Pages 8–9 Chain Groups Algebra I A group ( A , • ) is a set A together with a binary operation satisfying: Closure: for all α, β ∈ A , we have that α • β ∈ A . Associativity: so that for all α, β, γ ∈ A we have α • ( β • γ ) = ( α • β ) • γ . A has an identity element ω such that α + ω = α for all α ∈ A . If, in addition, • is commutative , we have that α • β = β • α for all α, β ∈ A , and we say the group ( A , • ) is abelian . Chains as Groups We can now recognize p -chains ( C p , +) as abelian groups. Chains as Vector Spaces If the complex K has n p p -simplices, then C p is (isomorphic to) the set of binary vectors of length n p , i.e., { 0 , 1 } n p , with the exclusive-or operation ⊕ . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 21 / 41

  52. Chains Pages 8–9 Chain Groups Algebra I A group ( A , • ) is a set A together with a binary operation satisfying: Closure: for all α, β ∈ A , we have that α • β ∈ A . Associativity: so that for all α, β, γ ∈ A we have α • ( β • γ ) = ( α • β ) • γ . A has an identity element ω such that α + ω = α for all α ∈ A . If, in addition, • is commutative , we have that α • β = β • α for all α, β ∈ A , and we say the group ( A , • ) is abelian . Chains as Groups We can now recognize p -chains ( C p , +) as abelian groups. Chains as Vector Spaces If the complex K has n p p -simplices, then C p is (isomorphic to) the set of binary vectors of length n p , i.e., { 0 , 1 } n p , with the exclusive-or operation ⊕ . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 21 / 41

  53. Chains Pages 8–9 Chain Groups Algebra I A group ( A , • ) is a set A together with a binary operation satisfying: Closure: for all α, β ∈ A , we have that α • β ∈ A . Associativity: so that for all α, β, γ ∈ A we have α • ( β • γ ) = ( α • β ) • γ . A has an identity element ω such that α + ω = α for all α ∈ A . If, in addition, • is commutative , we have that α • β = β • α for all α, β ∈ A , and we say the group ( A , • ) is abelian . Chains as Groups We can now recognize p -chains ( C p , +) as abelian groups. Chains as Vector Spaces If the complex K has n p p -simplices, then C p is (isomorphic to) the set of binary vectors of length n p , i.e., { 0 , 1 } n p , with the exclusive-or operation ⊕ . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 21 / 41

  54. Chains Pages 8–9 Chain Groups Algebra I A group ( A , • ) is a set A together with a binary operation satisfying: Closure: for all α, β ∈ A , we have that α • β ∈ A . Associativity: so that for all α, β, γ ∈ A we have α • ( β • γ ) = ( α • β ) • γ . A has an identity element ω such that α + ω = α for all α ∈ A . If, in addition, • is commutative , we have that α • β = β • α for all α, β ∈ A , and we say the group ( A , • ) is abelian . Chains as Groups We can now recognize p -chains ( C p , +) as abelian groups. Chains as Vector Spaces If the complex K has n p p -simplices, then C p is (isomorphic to) the set of binary vectors of length n p , i.e., { 0 , 1 } n p , with the exclusive-or operation ⊕ . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 21 / 41

  55. Chains Pages 8–9 Boundary of a Chain Linear Extensions Fix a p -simplex σ = [ v 0 , . . . , v p ] in the complex K . Recall that the boundary of σ is the collection of its proper faces, which we denoted by ∂σ . We can now express the boundary elements as a single ( p − 1)-chain p � ∂ p σ = [ v 0 , . . . , ˆ v i , . . . , v p ] , i =0 where ˆ v i indicates that v i is excluded in the corresponding face. Notice that we used the subscript to qualify the boundary operator as the one acting on the p -th chain group. For any p -chain c = � i a i σ i , its boundary is the ( p − 1)-chain �� � � ∂ p c = ∂ p a i σ i = a i ∂ p σ i . i i Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 22 / 41

  56. Chains Pages 8–9 Boundary of a Chain Linear Extensions Fix a p -simplex σ = [ v 0 , . . . , v p ] in the complex K . Recall that the boundary of σ is the collection of its proper faces, which we denoted by ∂σ . We can now express the boundary elements as a single ( p − 1)-chain p � ∂ p σ = [ v 0 , . . . , ˆ v i , . . . , v p ] , i =0 where ˆ v i indicates that v i is excluded in the corresponding face. Notice that we used the subscript to qualify the boundary operator as the one acting on the p -th chain group. For any p -chain c = � i a i σ i , its boundary is the ( p − 1)-chain �� � � ∂ p c = ∂ p a i σ i = a i ∂ p σ i . i i Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 22 / 41

  57. Chains Pages 8–9 Boundary of a Chain Linear Extensions Fix a p -simplex σ = [ v 0 , . . . , v p ] in the complex K . Recall that the boundary of σ is the collection of its proper faces, which we denoted by ∂σ . We can now express the boundary elements as a single ( p − 1)-chain p � ∂ p σ = [ v 0 , . . . , ˆ v i , . . . , v p ] , i =0 where ˆ v i indicates that v i is excluded in the corresponding face. Notice that we used the subscript to qualify the boundary operator as the one acting on the p -th chain group. For any p -chain c = � i a i σ i , its boundary is the ( p − 1)-chain �� � � ∂ p c = ∂ p a i σ i = a i ∂ p σ i . i i Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 22 / 41

  58. Chains Pages 8–9 Boundary of a Chain Linear Extensions Fix a p -simplex σ = [ v 0 , . . . , v p ] in the complex K . Recall that the boundary of σ is the collection of its proper faces, which we denoted by ∂σ . We can now express the boundary elements as a single ( p − 1)-chain p � ∂ p σ = [ v 0 , . . . , ˆ v i , . . . , v p ] , i =0 where ˆ v i indicates that v i is excluded in the corresponding face. Notice that we used the subscript to qualify the boundary operator as the one acting on the p -th chain group. For any p -chain c = � i a i σ i , its boundary is the ( p − 1)-chain �� � � ∂ p c = ∂ p a i σ i = a i ∂ p σ i . i i Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 22 / 41

  59. Chains Pages 8–9 Boundary of a Chain Linear Extensions Fix a p -simplex σ = [ v 0 , . . . , v p ] in the complex K . Recall that the boundary of σ is the collection of its proper faces, which we denoted by ∂σ . We can now express the boundary elements as a single ( p − 1)-chain p � ∂ p σ = [ v 0 , . . . , ˆ v i , . . . , v p ] , i =0 where ˆ v i indicates that v i is excluded in the corresponding face. Notice that we used the subscript to qualify the boundary operator as the one acting on the p -th chain group. For any p -chain c = � i a i σ i , its boundary is the ( p − 1)-chain �� � � ∂ p c = ∂ p a i σ i = a i ∂ p σ i . i i Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 22 / 41

  60. Chains Pages 8–9 The Chain Complex ∂ 3 ∂ 2 ∂ 1 ∂ 0 − → − → − → − → 0 Boundary Homomorphisms The boundary operator ∂ p commutes with the group operations. If c 1 and c 2 are p -chains, then: ∂ p ( c 1 + ( p ) c 2 ) = ∂ p c 1 + ( p − 1) ∂ p c 2 , where we qualify the addition operators on each side of the equation. This means that ∂ p induces a group homomorphism or a mapping between groups that preserves the group structures: ∂ p : C p → C p − 1 . We can arrange the chain groups into a chain complex, effectively replacing the geometric complex K with a series of algebraic modules. ∂ p +2 ∂ p +1 ∂ p ∂ p − 1 . . . − − → C p +1 − − → C p − → C p − 1 − − − → . . . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 23 / 41

  61. Chains Pages 8–9 The Chain Complex ∂ 3 ∂ 2 ∂ 1 ∂ 0 − → − → − → − → 0 Boundary Homomorphisms The boundary operator ∂ p commutes with the group operations. If c 1 and c 2 are p -chains, then: ∂ p ( c 1 + ( p ) c 2 ) = ∂ p c 1 + ( p − 1) ∂ p c 2 , where we qualify the addition operators on each side of the equation. This means that ∂ p induces a group homomorphism or a mapping between groups that preserves the group structures: ∂ p : C p → C p − 1 . We can arrange the chain groups into a chain complex, effectively replacing the geometric complex K with a series of algebraic modules. ∂ p +2 ∂ p +1 ∂ p ∂ p − 1 . . . − − → C p +1 − − → C p − → C p − 1 − − − → . . . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 23 / 41

  62. Chains Pages 8–9 The Chain Complex ∂ 3 ∂ 2 ∂ 1 ∂ 0 − → − → − → − → 0 Boundary Homomorphisms The boundary operator ∂ p commutes with the group operations. If c 1 and c 2 are p -chains, then: ∂ p ( c 1 + ( p ) c 2 ) = ∂ p c 1 + ( p − 1) ∂ p c 2 , where we qualify the addition operators on each side of the equation. This means that ∂ p induces a group homomorphism or a mapping between groups that preserves the group structures: ∂ p : C p → C p − 1 . We can arrange the chain groups into a chain complex, effectively replacing the geometric complex K with a series of algebraic modules. ∂ p +2 ∂ p +1 ∂ p ∂ p − 1 . . . − − → C p +1 − − → C p − → C p − 1 − − − → . . . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 23 / 41

  63. Chains Pages 8–9 The Chain Complex ∂ 3 ∂ 2 ∂ 1 ∂ 0 − → − → − → − → 0 Boundary Homomorphisms The boundary operator ∂ p commutes with the group operations. If c 1 and c 2 are p -chains, then: ∂ p ( c 1 + ( p ) c 2 ) = ∂ p c 1 + ( p − 1) ∂ p c 2 , where we qualify the addition operators on each side of the equation. This means that ∂ p induces a group homomorphism or a mapping between groups that preserves the group structures: ∂ p : C p → C p − 1 . We can arrange the chain groups into a chain complex, effectively replacing the geometric complex K with a series of algebraic modules. ∂ p +2 ∂ p +1 ∂ p ∂ p − 1 . . . − − → C p +1 − − → C p − → C p − 1 − − − → . . . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 23 / 41

  64. Chains Pages 8–9 But like .. what’s the point? Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 24 / 41

  65. Chains Pages 8–9 Boundary Matrices Chains Groups as Vector Spaces Let { σ i } i and { τ j } j denote the p -simplices and ( p − 1)-simplices of K . The boundary of a p -chain c = � i a i σ i is the ( p − 1)-chain �� � � � � � ∂ j , i ∂ p c = ∂ p a i σ i = a i ∂ p σ i = a i p τ j = b j τ j , i i i j j � a i ∂ j , i � , and ∂ j , i where b i = � is 1 if τ j ∈ ∂ p σ i and 0 otherwise. p p i With that, we can express the boundary operator ∂ p in matrix form.  ∂ 1 , n p  ∂ 1 , 1 ∂ 1 , 2 · · ·     b 1 a 1 p p p ∂ 2 , n p ∂ 2 , 1 ∂ 2 , 2 b 2 · · · a 2   p     p p   ∂ p c =  , ∂ p = , c = . .   . . .   ... .   . . . .     . . . . .        b n p − 1 ∂ n p − 1 , 0 ∂ n p − 1 , 2 ∂ n p − 1 , n p a n p · · · p p p Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 25 / 41

  66. Chains Pages 8–9 Boundary Matrices Chains Groups as Vector Spaces Let { σ i } i and { τ j } j denote the p -simplices and ( p − 1)-simplices of K . The boundary of a p -chain c = � i a i σ i is the ( p − 1)-chain �� � � � � � ∂ j , i ∂ p c = ∂ p a i σ i = a i ∂ p σ i = a i p τ j = b j τ j , i i i j j � a i ∂ j , i � , and ∂ j , i where b i = � is 1 if τ j ∈ ∂ p σ i and 0 otherwise. p p i With that, we can express the boundary operator ∂ p in matrix form.  ∂ 1 , n p  ∂ 1 , 1 ∂ 1 , 2 · · ·     b 1 a 1 p p p ∂ 2 , n p ∂ 2 , 1 ∂ 2 , 2 b 2 · · · a 2   p     p p   ∂ p c =  , ∂ p = , c = . .   . . .   ... .   . . . .     . . . . .        b n p − 1 ∂ n p − 1 , 0 ∂ n p − 1 , 2 ∂ n p − 1 , n p a n p · · · p p p Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 25 / 41

  67. Chains Pages 8–9 Boundary Matrices Chains Groups as Vector Spaces Let { σ i } i and { τ j } j denote the p -simplices and ( p − 1)-simplices of K . The boundary of a p -chain c = � i a i σ i is the ( p − 1)-chain �� � � � � � ∂ j , i ∂ p c = ∂ p a i σ i = a i ∂ p σ i = a i p τ j = b j τ j , i i i j j � a i ∂ j , i � , and ∂ j , i where b i = � is 1 if τ j ∈ ∂ p σ i and 0 otherwise. p p i With that, we can express the boundary operator ∂ p in matrix form.  ∂ 1 , n p  ∂ 1 , 1 ∂ 1 , 2 · · ·     b 1 a 1 p p p ∂ 2 , n p ∂ 2 , 1 ∂ 2 , 2 b 2 · · · a 2   p     p p   ∂ p c =  , ∂ p = , c = . .   . . .   ... .   . . . .     . . . . .        b n p − 1 ∂ n p − 1 , 0 ∂ n p − 1 , 2 ∂ n p − 1 , n p a n p · · · p p p Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 25 / 41

  68. Chains Pages 8–9 Boundaries and Cycles Which Boundaries are Useful? Consider the 1-chains on the torus to the right. We have a blue and a red loop. Also the boundary of the black triangle. Which of those help distinguish the torus from a sphere? Chains with No Boundary Any such chain is called a p -cycle. A p -cycle that arises as the boundary of a ( p + 1)-chain is a p -boundary. We need a way to count distinct p -cycles while ignoring all p -boundaries. Observe that ∂ p ◦ ∂ p +1 = 0. Figure from Wikipedia [18] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 26 / 41

  69. Chains Pages 8–9 Boundaries and Cycles Which Boundaries are Useful? Consider the 1-chains on the torus to the right. We have a blue and a red loop. Also the boundary of the black triangle. Which of those help distinguish the torus from a sphere? Chains with No Boundary We are particularly interested in p -chains c satisfying ∂ p c = ∅ . Any such chain is called a p -cycle. A p -cycle that arises as the boundary of a ( p + 1)-chain is a p -boundary. We need a way to count distinct p -cycles while ignoring all p -boundaries. Observe that ∂ p ◦ ∂ p +1 = 0. Figure from Wikipedia [18] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 26 / 41

  70. Chains Pages 8–9 Boundaries and Cycles Which Boundaries are Useful? Consider the 1-chains on the torus to the right. We have a blue and a red loop. Also the boundary of the black triangle. Which of those help distinguish the torus from a sphere? Chains with No Boundary We are particularly interested in p -chains c satisfying ∂ p c = ∅ . Any such chain is called a p -cycle. A p -cycle that arises as the boundary of a ( p + 1)-chain is a p -boundary. We need a way to count distinct p -cycles while ignoring all p -boundaries. Observe that ∂ p ◦ ∂ p +1 = 0. Figure from Wikipedia [18] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 26 / 41

  71. Chains Pages 8–9 Boundaries and Cycles Which Boundaries are Useful? Consider the 1-chains on the torus to the right. We have a blue and a red loop. Also the boundary of the black triangle. Which of those help distinguish the torus from a sphere? Chains with No Boundary We are particularly interested in p -chains c satisfying ∂ p c = ∅ . Any such chain is called a p -cycle. A p -cycle that arises as the boundary of a ( p + 1)-chain is a p -boundary. We need a way to count distinct p -cycles while ignoring all p -boundaries. Observe that ∂ p ◦ ∂ p +1 = 0. Figure from Wikipedia [18] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 26 / 41

  72. Chains Pages 8–9 Boundaries and Cycles Which Boundaries are Useful? Consider the 1-chains on the torus to the right. We have a blue and a red loop. Also the boundary of the black triangle. Which of those help distinguish the torus from a sphere? Chains with No Boundary We are particularly interested in p -chains c satisfying ∂ p c = ∅ . Any such chain is called a p -cycle. A p -cycle that arises as the boundary of a ( p + 1)-chain is a p -boundary. We need a way to count distinct p -cycles while ignoring all p -boundaries. Observe that ∂ p ◦ ∂ p +1 = 0. Figure from Wikipedia [18] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 26 / 41

  73. Chains Pages 8–9 Boundaries and Cycles Which Boundaries are Useful? Consider the 1-chains on the torus to the right. We have a blue and a red loop. Also the boundary of the black triangle. Which of those help distinguish the torus from a sphere? Chains with No Boundary We are particularly interested in p -chains c satisfying ∂ p c = ∅ . Any such chain is called a p -cycle. A p -cycle that arises as the boundary of a ( p + 1)-chain is a p -boundary. We need a way to count distinct p -cycles while ignoring all p -boundaries. Observe that ∂ p ◦ ∂ p +1 = 0. Figure from Wikipedia [18] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 26 / 41

  74. Chains Pages 8–9 Boundaries and Cycles Which Boundaries are Useful? Consider the 1-chains on the torus to the right. We have a blue and a red loop. Also the boundary of the black triangle. Which of those help distinguish the torus from a sphere? Chains with No Boundary We are particularly interested in p -chains c satisfying ∂ p c = ∅ . Any such chain is called a p -cycle. A p -cycle that arises as the boundary of a ( p + 1)-chain is a p -boundary. We need a way to count distinct p -cycles while ignoring all p -boundaries. Observe that ∂ p ◦ ∂ p +1 = 0. The fundamental lemma of homology! Figure from Wikipedia [18] Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 26 / 41

  75. Homology Page 10 Equivalence and Quotients Boundaries and Cycles as Subgroups Denote all p -cycles by Z p and all p -boundaries by B p . As the boundary map commutes with addition, Z p is a subgroup of C p . Likewise, B p is a subgroup of Z p . For any p -cycle α ∈ Z p and a p -boundary β , we get that α + β ∈ Z p . Algebra II We define an equivalence relation that identifies a pair of elements α, α ′ ∈ Z p whenever α ′ = α + β for some β ∈ B p . The equivalence relation partitions Z p into equivalence classes or cosets ; the coset [ α ] consists of all the elements identified with α . Then, the collection of cosets together with the addition operator give rise to the quotient group Z p / B p of the elements in Z p modulo the elements in B p . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 27 / 41

  76. Homology Page 10 Equivalence and Quotients Boundaries and Cycles as Subgroups Denote all p -cycles by Z p and all p -boundaries by B p . As the boundary map commutes with addition, Z p is a subgroup of C p . Likewise, B p is a subgroup of Z p . For any p -cycle α ∈ Z p and a p -boundary β , we get that α + β ∈ Z p . Algebra II We define an equivalence relation that identifies a pair of elements α, α ′ ∈ Z p whenever α ′ = α + β for some β ∈ B p . The equivalence relation partitions Z p into equivalence classes or cosets ; the coset [ α ] consists of all the elements identified with α . Then, the collection of cosets together with the addition operator give rise to the quotient group Z p / B p of the elements in Z p modulo the elements in B p . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 27 / 41

  77. Homology Page 10 Equivalence and Quotients Boundaries and Cycles as Subgroups Denote all p -cycles by Z p and all p -boundaries by B p . As the boundary map commutes with addition, Z p is a subgroup of C p . Likewise, B p is a subgroup of Z p . For any p -cycle α ∈ Z p and a p -boundary β , we get that α + β ∈ Z p . Algebra II We define an equivalence relation that identifies a pair of elements α, α ′ ∈ Z p whenever α ′ = α + β for some β ∈ B p . The equivalence relation partitions Z p into equivalence classes or cosets ; the coset [ α ] consists of all the elements identified with α . Then, the collection of cosets together with the addition operator give rise to the quotient group Z p / B p of the elements in Z p modulo the elements in B p . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 27 / 41

  78. Homology Page 10 Equivalence and Quotients Boundaries and Cycles as Subgroups Denote all p -cycles by Z p and all p -boundaries by B p . As the boundary map commutes with addition, Z p is a subgroup of C p . Likewise, B p is a subgroup of Z p . For any p -cycle α ∈ Z p and a p -boundary β , we get that α + β ∈ Z p . Algebra II We define an equivalence relation that identifies a pair of elements α, α ′ ∈ Z p whenever α ′ = α + β for some β ∈ B p . The equivalence relation partitions Z p into equivalence classes or cosets ; the coset [ α ] consists of all the elements identified with α . Then, the collection of cosets together with the addition operator give rise to the quotient group Z p / B p of the elements in Z p modulo the elements in B p . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 27 / 41

  79. Homology Page 10 Equivalence and Quotients Boundaries and Cycles as Subgroups Denote all p -cycles by Z p and all p -boundaries by B p . As the boundary map commutes with addition, Z p is a subgroup of C p . Likewise, B p is a subgroup of Z p . For any p -cycle α ∈ Z p and a p -boundary β , we get that α + β ∈ Z p . Algebra II We define an equivalence relation that identifies a pair of elements α, α ′ ∈ Z p whenever α ′ = α + β for some β ∈ B p . The equivalence relation partitions Z p into equivalence classes or cosets ; the coset [ α ] consists of all the elements identified with α . Then, the collection of cosets together with the addition operator give rise to the quotient group Z p / B p of the elements in Z p modulo the elements in B p . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 27 / 41

  80. Homology Page 10 Equivalence and Quotients Boundaries and Cycles as Subgroups Denote all p -cycles by Z p and all p -boundaries by B p . As the boundary map commutes with addition, Z p is a subgroup of C p . Likewise, B p is a subgroup of Z p . For any p -cycle α ∈ Z p and a p -boundary β , we get that α + β ∈ Z p . Algebra II We define an equivalence relation that identifies a pair of elements α, α ′ ∈ Z p whenever α ′ = α + β for some β ∈ B p . The equivalence relation partitions Z p into equivalence classes or cosets ; the coset [ α ] consists of all the elements identified with α . Then, the collection of cosets together with the addition operator give rise to the quotient group Z p / B p of the elements in Z p modulo the elements in B p . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 27 / 41

  81. Homology Page 10 Equivalence and Quotients Boundaries and Cycles as Subgroups Denote all p -cycles by Z p and all p -boundaries by B p . As the boundary map commutes with addition, Z p is a subgroup of C p . Likewise, B p is a subgroup of Z p . For any p -cycle α ∈ Z p and a p -boundary β , we get that α + β ∈ Z p . Algebra II We define an equivalence relation that identifies a pair of elements α, α ′ ∈ Z p whenever α ′ = α + β for some β ∈ B p . The equivalence relation partitions Z p into equivalence classes or cosets ; the coset [ α ] consists of all the elements identified with α . Then, the collection of cosets together with the addition operator give rise to the quotient group Z p / B p of the elements in Z p modulo the elements in B p . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 27 / 41

  82. Homology Page 10 Homology Algebra III Take a group ( A , • ). The order of the group is the cardinality of A . The rank of the group is the cardinality of a minimal generator . For a set of binary vectors, such as C p or Z p The order is the number of distinct binary vectors. The rank is the number basis vectors that span the entire set. Homology Groups and Betti Numbers We can now defined the p -th homology group as H p = Z p / B p . The rank of H p is known as the p-th Betti number β p β p = rank H p = rank Z p − rank B p . Ahmed Abdelkader (CS@UMD) Introduction to Computational Topology May 7th, 2020 28 / 41

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