Cohomology Seminar Algorithms Jari de Kroon Eindhoven University of Technology May 22, 2018
Content ◮ Motivation ◮ Cohomology groups ◮ Universal Coefficient Theorem ◮ Application [SMV11]
Motivation ◮ Distinguish topological spaces ◮ Algebraic dual of homology
P-cochain Recall: For simplicial complex K, a p-chain is a formal sum c = � n i σ i of p -simplices. C p = C p ( K ; G ) is the group of p-chains closed under addition, with constants n i ∈ G .
P-cochain Recall: For simplicial complex K, a p-chain is a formal sum c = � n i σ i of p -simplices. C p = C p ( K ; G ) is the group of p-chains closed under addition, with constants n i ∈ G . Define a p-cochain as a homomorphism ϕ : C p → G . These form the group of p-cochains , C p = Hom ( C p , G ).
P-cochain Recall: For simplicial complex K, a p-chain is a formal sum c = � n i σ i of p -simplices. C p = C p ( K ; G ) is the group of p-chains closed under addition, with constants n i ∈ G . Define a p-cochain as a homomorphism ϕ : C p → G . These form the group of p-cochains , C p = Hom ( C p , G ). Example: Take G = Z 2 , then given a p -chain c ∈ C p , the cochain maps c to 0 or 1. Intuitively can be seen as functions from p -simplices to 0 or 1, as the p -simplices form the basis of C p .
P-cochain The basis of C p are the p -simplices σ 1 , σ 2 , .., σ n . The basis of C p are the cochains f 1 , f 2 , ..., f n such that f i ( σ j ) = δ i , j , the Kronecker delta that is 1 if i = j .
P-cochain The basis of C p are the p -simplices σ 1 , σ 2 , .., σ n . The basis of C p are the cochains f 1 , f 2 , ..., f n such that f i ( σ j ) = δ i , j , the Kronecker delta that is 1 if i = j . Example: Take � 1 , if x = v 0 or x = v 1 ϕ ( x ) = 0 , otherwise
P-cochain The basis of C p are the p -simplices σ 1 , σ 2 , .., σ n . The basis of C p are the cochains f 1 , f 2 , ..., f n such that f i ( σ j ) = δ i , j , the Kronecker delta that is 1 if i = j . Example: Take � 1 , if x = v 0 or x = v 1 ϕ ( x ) = 0 , otherwise Then ϕ ( x ) = f 0 ( x ) + f 1 ( x ).
Coboundary map Recall: The boundary operator is a homomorphism ∂ p : C p → C p − 1 . For p -simplex σ = [ v 0 , .., v p ], we had p � ( − 1) i [ v 0 , .., ˆ ∂ p σ = v i , .., v p ] i =0
Coboundary map Recall: The boundary operator is a homomorphism ∂ p : C p → C p − 1 . For p -simplex σ = [ v 0 , .., v p ], we had p � ( − 1) i [ v 0 , .., ˆ ∂ p σ = v i , .., v p ] i =0 Define the coboundary operator as δ p : C p → C p +1 . Then given a p -cochain ϕ ∈ C p = Hom ( C i , G ), we now define p +1 � δ p ϕ ([ v 0 , .., v p +1 ]) = ( − 1) i ϕ ([ v 0 , .., ˆ v i , .., v p +1 ]) i =0
Example Take G = Z 2 . Let ϕ be a 0-cochain defined as � 1 , if x = v 0 ϕ ( x ) = 0 , otherwise Then δ 0 ϕ is a cochain on the edges. v 0 v 1 δ 0 ϕ ([ v 0 v 1 ]) v 2
Example Take G = Z 2 . Let ϕ be a 0-cochain defined as � 1 , if x = v 0 ϕ ( x ) = 0 , otherwise Then δ 0 ϕ is a cochain on the edges. v 0 v 1 δ 0 ϕ ([ v 0 v 1 ]) = ϕ ([ v 1 ]) − ϕ ([ v 0 ]) = 0 − 1 v 2
Example Take G = Z 2 . Let ϕ be a 0-cochain defined as � 1 , if x = v 0 ϕ ( x ) = 0 , otherwise Then δ 0 ϕ is a cochain on the edges. v 0 v 1 δ 0 ϕ ([ v 0 v 1 ]) = ϕ ([ v 1 ]) − ϕ ([ v 0 ]) = 0 − 1 δ 0 ϕ ([ v 1 v 2 ]) v 2
Example Take G = Z 2 . Let ϕ be a 0-cochain defined as � 1 , if x = v 0 ϕ ( x ) = 0 , otherwise Then δ 0 ϕ is a cochain on the edges. v 0 v 1 δ 0 ϕ ([ v 0 v 1 ]) = ϕ ([ v 1 ]) − ϕ ([ v 0 ]) = 0 − 1 δ 0 ϕ ([ v 1 v 2 ]) = ϕ ([ v 2 ]) − ϕ ([ v 1 ]) = 0 − 0 v 2
Example Take G = Z 2 . Let ϕ be a 0-cochain defined as � 1 , if x = v 0 ϕ ( x ) = 0 , otherwise Then δ 0 ϕ is a cochain on the edges. v 0 v 1 δ 0 ϕ ([ v 0 v 1 ]) = ϕ ([ v 1 ]) − ϕ ([ v 0 ]) = 0 − 1 δ 0 ϕ ([ v 1 v 2 ]) = ϕ ([ v 2 ]) − ϕ ([ v 1 ]) = 0 − 0 v 2 δ 0 ϕ ([ v 2 v 0 ])
Example Take G = Z 2 . Let ϕ be a 0-cochain defined as � 1 , if x = v 0 ϕ ( x ) = 0 , otherwise Then δ 0 ϕ is a cochain on the edges. v 0 v 1 δ 0 ϕ ([ v 0 v 1 ]) = ϕ ([ v 1 ]) − ϕ ([ v 0 ]) = 0 − 1 δ 0 ϕ ([ v 1 v 2 ]) = ϕ ([ v 2 ]) − ϕ ([ v 1 ]) = 0 − 0 v 2 δ 0 ϕ ([ v 2 v 0 ]) = ϕ ([ v 0 ]) − ϕ ([ v 2 ]) = 1 − 0
Example Take G = Z 2 . Let ϕ be a 0-cochain defined as � 1 , if x = v 0 ϕ ( x ) = 0 , otherwise Then δ 0 ϕ is a cochain on the edges. v 0 v 1 δ 0 ϕ ([ v 0 v 1 ]) = ϕ ([ v 1 ]) − ϕ ([ v 0 ]) = 0 − 1 δ 0 ϕ ([ v 1 v 2 ]) = ϕ ([ v 2 ]) − ϕ ([ v 1 ]) = 0 − 0 v 2 δ 0 ϕ ([ v 2 v 0 ]) = ϕ ([ v 0 ]) − ϕ ([ v 2 ]) = 1 − 0
Example Then δ 1 ( δ 0 ϕ ) is a cochain on the triangles. δ 1 ( δ 0 ϕ )([ v 0 v 1 v 2 ]) v 0 v 1 v 2
Example Then δ 1 ( δ 0 ϕ ) is a cochain on the triangles. δ 1 ( δ 0 ϕ )([ v 0 v 1 v 2 ]) v 0 v 1 = δ 0 ϕ ([ v 1 v 2 ]) − δ 0 ϕ ([ v 0 v 2 ]) + δ 0 ϕ ([ v 0 v 1 ]) = 0 − 1 + 1 = 0 v 2
Example Then δ 1 ( δ 0 ϕ ) is a cochain on the triangles. δ 1 ( δ 0 ϕ )([ v 0 v 1 v 2 ]) v 0 v 1 = δ 0 ϕ ([ v 1 v 2 ]) − δ 0 ϕ ([ v 0 v 2 ]) + δ 0 ϕ ([ v 0 v 1 ]) = 0 − 1 + 1 = 0 v 2 No coincidence, in general δ p +1 δ p = 0.
Example Then δ 1 ( δ 0 ϕ ) is a cochain on the triangles. δ 1 ( δ 0 ϕ )([ v 0 v 1 v 2 ]) v 0 v 1 = δ 0 ϕ ([ v 1 v 2 ]) − δ 0 ϕ ([ v 0 v 2 ]) + δ 0 ϕ ([ v 0 v 1 ]) = 0 − 1 + 1 = 0 v 2 No coincidence, in general δ p +1 δ p = 0. In Z 2 , an edge is evaluated to 1 iff vertices are labeled differently. A triangle is evaluated to 1 iff an odd number of edges evaluate to 1.
Homology groups Recall: Group of p-cycles Z p is the kernel of ∂ p . Group of p-boundaries B p is the image of ∂ p +1 . The p-th homology group H p = Z p / B p .
Cohomology groups In cohomology we define: Group of p-cocycles Z p is the kernel of δ p . Group of p-coboundaries B p is the image of δ p − 1 . The p-th cohomology group H p = Z p / B p .
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − −
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − rank H 0 =
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − rank H 0 = 1 (connected components)
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − 0-cocycles := labeling of vertices such that all edges evaluate to zero.
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − 0-cocycles := labeling of vertices such that all edges evaluate to zero. ϕ ( x ) = 1
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − 0-cocycles := labeling of vertices such that all edges evaluate to zero. ϕ ( x ) = 1 Zero-homomorphism is trivial solution, not counted.
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − 0-cocycles := labeling of vertices such that all edges evaluate to zero. ϕ ( x ) = 1 Zero-homomorphism is trivial solution, not counted. 0-coboundaries do not exist, as there is no C − 1 .
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − 0-cocycles := labeling of vertices such that all edges evaluate to zero. ϕ ( x ) = 1 Zero-homomorphism is trivial solution, not counted. 0-coboundaries do not exist, as there is no C − 1 . So rank H 0 = 1
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − rank H 1 =
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − rank H 1 = 1 (1-dimensional hole)
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − 1-cocycles := labeling of edges such that all triangles evaluate to zero.
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − 1-cocycles := labeling of edges such that all triangles evaluate to zero. Every triangle incident to even number of edges evaluating to one.
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − 1-cocycles := labeling of edges such that all triangles evaluate to zero. Every triangle incident to even number of edges evaluating to one. 1-cocycle looks like picket fence (red).
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − 1-cocycles := labeling of edges such that all triangles evaluate to zero. Every triangle incident to even number of edges evaluating to one. 1-cocycle looks like picket fence (red). 1-coboundary := coboundary of orange vertices are red edges.
Ex. Homology vs cohomology groups 0 → C 0 δ 0 → C 1 δ 1 → C 2 δ 2 → 0, G = Z 2 − − − 1-cocycles := labeling of edges such that all triangles evaluate to zero. Every triangle incident to even number of edges evaluating to one. 1-cocycle looks like picket fence (red). 1-coboundary := coboundary of orange vertices are red edges. Since H p = Z p / B p , we want 1-cocycles that is not also a 1-coboundary.
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