Cohomology Combinatorial Cellular & Abstract Eilenberg-Steenrod Ulrik Buchholtz and Favonia 1
Cohomology Groups { mappings from holes in a space } 2
Cohomology Groups { mappings from holes in a space } Cellular Axiomatic Eilenberg-Steenrod cohomology for CW complexes cohomology 2
Cohomology Groups { mappings from holes in a space } Cellular Axiomatic Eilenberg-Steenrod cohomology for CW complexes cohomology Dream: prove they are the same! 2
CW complexes inductively-de � ined spaces 3
CW complexes inductively-de � ined spaces points 3
CW complexes inductively-de � ined spaces points lines 3
CW complexes inductively-de � ined spaces points lines faces 3
CW complexes inductively-de � ined spaces points lines faces (and more...) 3
CW complexes inductively-de � ined spaces points lines faces (and more...) Speci � ication: cells and how they a � ach 3
CW complexes Sets of cells: A n 4
CW complexes Sets of cells: A n A � aching: α n+1 : A n+1 × S n → X n X n is the construction up to dim. n 4
CW complexes Sets of cells: A n A � aching: α n+1 : A n+1 × S n → X n X n is the construction up to dim. n X n 4
CW complexes Sets of cells: A n A � aching: α n+1 : A n+1 × S n → X n X n is the construction up to dim. n a : A n+1 X n 4
CW complexes Sets of cells: A n A � aching: α n+1 : A n+1 × S n → X n X n is the construction up to dim. n a : A n+1 S n α n+1 (a,-) X n 4
CW complexes Sets of cells: A n A � aching: α n+1 : A n+1 × S n → X n X n is the construction up to dim. n X 0 := A 0 a : A n+1 S n α n+1 (a,-) X n 4
CW complexes Sets of cells: A n A � aching: α n+1 : A n+1 × S n → X n X n is the construction up to dim. n X 0 := A 0 X n+1 := a : A n+1 A n+1 ×S n A n+1 S n α n+1 (a,-) X n α n+1 X n X n+1 4
Cellular Cohomology { mappings from holes in a space } 5
Cellular Cohomology { mappings from holes in a space } Cellular Homology { holes in a space } 5
Cellular Cohomology { mappings from holes in a space } dualize Cellular Homology { holes in a space } 5
One-Dimensional Holes* { elements of Z[A 1 ] forming cycles } holes a d c b e f g *Holes are cycles in the classical homology theory 6
One-Dimensional Holes* { elements of Z[A 1 ] forming cycles } holes a d a + b + c c b e f g *Holes are cycles in the classical homology theory 6
One-Dimensional Holes* { elements of Z[A 1 ] forming cycles } holes a d a + b + c c b - a - b - c e f g *Holes are cycles in the classical homology theory 6
One-Dimensional Holes* { elements of Z[A 1 ] forming cycles } holes a d a + b + c c b - a - b - c e f a + b + c + e + g + f … g *Holes are cycles in the classical homology theory 6
One-Dimensional Holes { elements of Z[A 1 ] forming cycles } a boundary function ∂ a ∂ ( ) = y - x c b x y 7
One-Dimensional Holes { elements of Z[A 1 ] forming cycles } a boundary function ∂ a ∂ ( ) = y - x c b x y set of holes = kernel of ∂ 7
One-Dimensional Holes { elements of Z[A 1 ] forming cycles } x a boundary function ∂ y a ∂ ( ) = y - x c b x y z set of holes = kernel of ∂ 7
One-Dimensional Holes { elements of Z[A 1 ] forming cycles } x a boundary function ∂ y a ∂ ( ) = y - x c b x y ∂ (a+b+c) = (y - x) + (z - y) z + (x - z) = 0 set of holes = kernel of ∂ 7
First Homology Groups { un � illed one-dimensional holes } a p c b 8
First Homology Groups { un � illed one-dimensional holes } a 2-dim. boundary function ∂ 2 p a c b ∂ 2 ( ) = a + b + c p c b 8
First Homology Groups { un � illed one-dimensional holes } a 2-dim. boundary function ∂ 2 p a c b ∂ 2 ( ) = a + b + c p c b � illed holes = image of ∂ 2 8
First Homology Groups { un � illed one-dimensional holes } a 2-dim. boundary function ∂ 2 p a c b ∂ 2 ( ) = a + b + c p c b � illed holes = image of ∂ 2 H 1 (X) := kernel of ∂ 1 / image of ∂ 2 (un � illed (all holes) ( � illed holes) holes) 8
Homology Groups { un � illed holes } C n := Z[A n ] formal sums of cells (chains) 9
Homology Groups { un � illed holes } C n := Z[A n ] formal sums of cells (chains) ∂ n+2 ∂ n+1 ∂ n ∂ n-1 ⋯ → C n+2 → C n+1 → C n → C n-1 → C n-2 → ⋯ 9
Homology Groups { un � illed holes } C n := Z[A n ] formal sums of cells (chains) ∂ n+2 ∂ n+1 ∂ n ∂ n-1 ⋯ → C n+2 → C n+1 → C n → C n-1 → C n-2 → ⋯ H n (X) := kernel of ∂ n / image of ∂ n+1 9
Cohomology Groups ∂ n+2 ∂ n+1 ∂ n ∂ n-1 ⋯ → C n+2 → C n+1 → C n → C n-1 → C n-2 → ⋯ Dualize by Hom(—, G). Let C n = Hom(C n , G). δ n+2 δ n+1 δ n δ n-1 ⋯ ← C n+2 ← C n+1 ← C n ← C n-1 ← C n-2 ← ⋯ 10
Cohomology Groups ∂ n+2 ∂ n+1 ∂ n ∂ n-1 ⋯ → C n+2 → C n+1 → C n → C n-1 → C n-2 → ⋯ Dualize by Hom(—, G). Let C n = Hom(C n , G). δ n+2 δ n+1 δ n δ n-1 ⋯ ← C n+2 ← C n+1 ← C n ← C n-1 ← C n-2 ← ⋯ H n (X; G) := kernel of δ n+1 / image of δ n 10
Higher-Dim. Boundary a p c b a ∂ 2 ( ) = a + b + c p c b How to compute the coe � icients from α 2 ? 11
Higher-Dim. Boundary a p c b a a a α 2 (p,—) identify squash points other loops coe � icient = winding number of this map 12
Higher-Dim. Boundary α 2 (p,—) X n /X n-1 ≃⋁ S n S n S n X n α n+1 (p,—) identify squash lower structs. coe � icient = degree of this map 13
Higher-Dim. Boundary α 2 (p,—) X n /X n-1 ≃⋁ S n S n S n X n α n+1 (p,—) identify squash lower structs. coe � icient = degree of this map - squashing needs decidable equality - linear sum needs closure- � initeness 13
Higher-Dim. Boundary S n A n ×S n-1 A n+1 ×S n A n A n+1 X n-1 X n X n+1 X n /X n-1 ≃⋁ S n 1 S n 14
Cohomology Groups { mappings from holes in a space } Cellular Axiomatic Eilenberg-Steenrod cohomology for H n (X; G) CW-complexes cohomology Dream: prove they are the same! 15
Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups * Ordinary, reduced cohomology theory 16
Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups 1. h n+1 (susp(X)) ≃ h n (X), natural in X * Ordinary, reduced cohomology theory 16
Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups 1. h n+1 (susp(X)) ≃ h n (X), natural in X 2. f A B 1 Cof f * Ordinary, reduced cohomology theory 16
Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups 1. h n+1 (susp(X)) ≃ h n (X), natural in X h n (A) h n (A) h n (B) 2. f A B exact! Cof f h n (Cof f ) 1 * Ordinary, reduced cohomology theory 16
Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups 1. h n+1 (susp(X)) ≃ h n (X), natural in X 3. h n ( ⋁ i X i ) ≃ ∏ i h n (X i ) h n (A) h n (A) h n (B) 2. f if the index type A B exact! satis � ies set-level AC Cof f h n (Cof f ) 1 * Ordinary, reduced cohomology theory 16
Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups 1. h n+1 (susp(X)) ≃ h n (X), natural in X 3. h n ( ⋁ i X i ) ≃ ∏ i h n (X i ) h n (A) h n (A) h n (B) 2. f if the index type A B exact! satis � ies set-level AC Cof f h n (Cof f ) 4. h n (2) trivial for n ≠ 0 1 * Ordinary, reduced cohomology theory 16
Cohomology Groups { mappings from holes in a space } Cellular Axiomatic Eilenberg-Steenrod cohomology for H n (X; G) h n (X) CW-complexes cohomology Dream: prove they are the same! 17
Our Dream ? h n (X) ≃ H n (X; h 0 (2)) 18
Our Dream ? h n (X) ≃ H n (X; h 0 (2)) := ker( δ n+1 )/im( δ n ) 18
Our Dream ? h n (X) ≃ H n (X; h 0 (2)) ≃ ? := ker( δ ' n+1 )/im( δ ' n ) ker( δ n+1 )/im( δ n ) 18
Our Dream ? h n (X) ≃ H n (X; h 0 (2)) ≃ ? := ker( δ ' n+1 )/im( δ ' n ) ker( δ n+1 )/im( δ n ) 1. Find δ ' such that h n (X) ≃ ker( δ ' n+1 )/im( δ ' n ) done and fully mechanized in Agda 2. Show δ and δ ' are equivalent domains and codomains are isomorphic commutativity in progress 18
Our Dream: Step 1 (done!) For any pointed CW-complex X where 1. all cell sets A n satisfy set-level AC and 2. the point of A 0 is separable (pt = x is decidable) there exist homomorphisms δ ' δ ' n+2 δ ' n+1 δ ' n δ ' n-1 ⋯ ← D n+2 ← D n+1 ← D n ← D n-1 ← D n-2 ← ⋯ such that h n (X) ≃ kernel of δ ' n+1 / image of δ ' n 19
Important Lemmas for Step 1 Long exact sequenses h n (A) h n (B) f A B n++ Cof f h n (Cof f ) 1 20
Important Lemmas for Step 1 Long exact sequenses h n (A) h n (B) f A B n++ Cof f h n (Cof f ) 1 Wedges of cells h m (X n /X n-1 ) ≃ hom(Z[A n ], h 0 (2)) when m = n or trivial otherwise h m (X 0 ) ≃ hom(Z[A 0 \{pt}], h 0 (2)) when m = 0 or trivial otherwise 20
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