Homology? Its Mickey Mouse! So whats the big deal? Ron Umble, - PowerPoint PPT Presentation

Cellular Approximation Theorem There is a map ∆ homotopic to ∆ G s.t. ∆ ( X ) is a union of cells of X Example: ∆ G : I → I × I is homotopic to  � �  � � 0 , 1 0 , 1 ( 0 , 2 t ) t ∈ ( 2 t , 0 ) t ∈   2 2 f ( t ) = and g ( t ) = � 1 � � 1 �   ( 2 t , 1 ) t ∈ ( 1 , 2 t ) t ∈ 2 , 1 2 , 1 (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 16 / 37

Cellular Approximation Theorem There is a map ∆ homotopic to ∆ G s.t. ∆ ( X ) is a union of cells of X Example: ∆ G : I → I × I is homotopic to  � �  � � 0 , 1 0 , 1 ( 0 , 2 t ) t ∈ ( 2 t , 0 ) t ∈   2 2 f ( t ) = and g ( t ) = � 1 � � 1 �   ( 2 t , 1 ) t ∈ ( 1 , 2 t ) t ∈ 2 , 1 2 , 1 f and g are diagonal approximations of ∆ G (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 16 / 37

Cellular Approximation Theorem There is a map ∆ homotopic to ∆ G s.t. ∆ ( X ) is a union of cells of X Example: ∆ G : I → I × I is homotopic to  � �  � � 0 , 1 0 , 1 ( 0 , 2 t ) t ∈ ( 2 t , 0 ) t ∈   2 2 f ( t ) = and g ( t ) = � 1 � � 1 �   ( 2 t , 1 ) t ∈ ( 1 , 2 t ) t ∈ 2 , 1 2 , 1 f and g are diagonal approximations of ∆ G Standard choice is ∆ : = f ∆ ( I ) = 0 × I + I × 1 (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 16 / 37

Diagonal approximations A diagonal approximation ∆ : X → X × X preserves Cell structure: ∆ ( c ) ⊆ c × c (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 17 / 37

Diagonal approximations A diagonal approximation ∆ : X → X × X preserves Cell structure: ∆ ( c ) ⊆ c × c Dimension: If dim c = k , then dim ∆ ( c ) = k (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 17 / 37

Diagonal approximations A diagonal approximation ∆ : X → X × X preserves Cell structure: ∆ ( c ) ⊆ c × c Dimension: If dim c = k , then dim ∆ ( c ) = k Wedge products: ∆ ( X ∨ Y ) = ∆ ( X ) ∨ ∆ ( Y ) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 17 / 37

Diagonal approximations A diagonal approximation ∆ : X → X × X preserves Cell structure: ∆ ( c ) ⊆ c × c Dimension: If dim c = k , then dim ∆ ( c ) = k Wedge products: ∆ ( X ∨ Y ) = ∆ ( X ) ∨ ∆ ( Y ) Cartesian products: ∆ ( X × Y ) = ∆ ( X ) × ∆ ( Y ) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 17 / 37

Diagonal approximations A diagonal approximation ∆ : X → X × X preserves Cell structure: ∆ ( c ) ⊆ c × c Dimension: If dim c = k , then dim ∆ ( c ) = k Wedge products: ∆ ( X ∨ Y ) = ∆ ( X ) ∨ ∆ ( Y ) Cartesian products: ∆ ( X × Y ) = ∆ ( X ) × ∆ ( Y ) � I 2 � = ∆ ( I ) × ∆ ( I ) ∆ Example: = ( 0 × I + I × 1 ) × ( 0 × I + I × 1 ) 00 × I 2 + 0 I × I 1 + I 0 × 1 I + I 2 × 11 = (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 17 / 37

Dan Kravatz’s diagonal approximation on a polygon Given n -gon G , n ≥ 1 , choose arbitrary vertices v and v � (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 18 / 37

Dan Kravatz’s diagonal approximation on a polygon Given n -gon G , n ≥ 1 , choose arbitrary vertices v and v � Edges { e 1 , . . . , e k } and { e k + 1 , . . . , e n } form paths from v to v � (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 18 / 37

Dan Kravatz’s diagonal approximation on a polygon Given n -gon G , n ≥ 1 , choose arbitrary vertices v and v � Edges { e 1 , . . . , e k } and { e k + 1 , . . . , e n } form paths from v to v � Theorem (Kravatz 2008): There is a diagonal approximation k n ∑ ∑ ∆ G = ( e 1 + · · · + e i − 1 ) × e i + ( e k + 1 + · · · + e j − 1 ) × e j i = 2 j = k + 2 + v × G + G × v � (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 18 / 37

Example Think of the sphere S 2 as a 1-gon with its edge identified ( v = v � ) ∆ S 2 = v × S 2 + S 2 × v (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 19 / 37

Example Think of the torus T as a square with horizontal edges a identified and vertical edges b identified ∆ T = v × T + T × v + a × b + b × a (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 20 / 37

Example Think of Mickey Mouse M as the wedge of a 1-gon S 2 with intervals a and b ; the edge of S 2 and the endpoints of a and b are identified � S 2 ∨ S 1 ∨ S 1 � = ∆ S 2 + ∆ S 1 + ∆ S 1 ∆ M ∆ = � � + ( v × a + a × v ) + ( v × b + b × v ) v × S 2 + S 2 × v = � � + � � × v S 2 + a + b S 2 + a + b = v × = v × M + M × v (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 21 / 37

Example Think of the pinched sphere t 1 in ∂ ( UN ) as a 2-gon with vertices identified first, then edges identified ∆ t 1 = v × t 1 + t 1 × v (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 22 / 37

Example Think of the torus t � 1 in ∂ ( LN ) as a square with horizontal edges identified a and vertical edges b identified ∆ t � 1 = v × t � 1 + t � 1 × v + a × b + b × a As it stands, ∆ T � = ∆ M � T � M and ∆ t 1 � = ∆ t � 1 � UN � LN (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 23 / 37

Example Think of the torus t � 1 in ∂ ( LN ) as a square with horizontal edges identified a and vertical edges b identified ∆ t � 1 = v × t � 1 + t � 1 × v + a × b + b × a As it stands, ∆ T � = ∆ M � T � M and ∆ t 1 � = ∆ t � 1 � UN � LN Are these diagonal approximations irreconcilably different? (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 23 / 37

Example Think of the torus t � 1 in ∂ ( LN ) as a square with horizontal edges identified a and vertical edges b identified ∆ t � 1 = v × t � 1 + t � 1 × v + a × b + b × a As it stands, ∆ T � = ∆ M � T � M and ∆ t 1 � = ∆ t � 1 � UN � LN Are these diagonal approximations irreconcilably different? We need homology (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 23 / 37

Cellular chains of X Elements of Z 2 -vector space C ( X ) with basis { cells of X } (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

Cellular chains of X Elements of Z 2 -vector space C ( X ) with basis { cells of X } Chains are finite sums of cells of X (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

Cellular chains of X Elements of Z 2 -vector space C ( X ) with basis { cells of X } Chains are finite sums of cells of X C ( 8 ) = { v , a , b , v + a , v + b , a + b , v + a + b } (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

Cellular chains of X Elements of Z 2 -vector space C ( X ) with basis { cells of X } Chains are finite sums of cells of X C ( 8 ) = { v , a , b , v + a , v + b , a + b , v + a + b } � v , a , b , S 2 � C ( M ) has basis (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

Cellular chains of X Elements of Z 2 -vector space C ( X ) with basis { cells of X } Chains are finite sums of cells of X C ( 8 ) = { v , a , b , v + a , v + b , a + b , v + a + b } � v , a , b , S 2 � C ( M ) has basis C ( T ) has basis { v , a , b , T } (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

Cellular chains of X Elements of Z 2 -vector space C ( X ) with basis { cells of X } Chains are finite sums of cells of X C ( 8 ) = { v , a , b , v + a , v + b , a + b , v + a + b } � v , a , b , S 2 � C ( M ) has basis C ( T ) has basis { v , a , b , T } Note that C ( T ) ≈ C ( M ) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

Cellular chains of X Elements of Z 2 -vector space C ( X ) with basis { cells of X } Chains are finite sums of cells of X C ( 8 ) = { v , a , b , v + a , v + b , a + b , v + a + b } � v , a , b , S 2 � C ( M ) has basis C ( T ) has basis { v , a , b , T } Note that C ( T ) ≈ C ( M ) C ( UN ) has basis { v , a , b , s , t 1 , t 2 , p , q } (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

Cellular chains of X Elements of Z 2 -vector space C ( X ) with basis { cells of X } Chains are finite sums of cells of X C ( 8 ) = { v , a , b , v + a , v + b , a + b , v + a + b } � v , a , b , S 2 � C ( M ) has basis C ( T ) has basis { v , a , b , T } Note that C ( T ) ≈ C ( M ) C ( UN ) has basis { v , a , b , s , t 1 , t 2 , p , q } C ( LN ) has basis { v , a , b , s , t � 1 , t � 2 , p , q � } (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

Cellular chains of X Elements of Z 2 -vector space C ( X ) with basis { cells of X } Chains are finite sums of cells of X C ( 8 ) = { v , a , b , v + a , v + b , a + b , v + a + b } � v , a , b , S 2 � C ( M ) has basis C ( T ) has basis { v , a , b , T } Note that C ( T ) ≈ C ( M ) C ( UN ) has basis { v , a , b , s , t 1 , t 2 , p , q } C ( LN ) has basis { v , a , b , s , t � 1 , t � 2 , p , q � } Note that C ( UN ) ≈ C ( LN ) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

Geometric boundary Geometric boundary of a cell c is either empty or a union of (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

Geometric boundary Geometric boundary of a cell c is either empty or a union of Vertices v (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

Geometric boundary Geometric boundary of a cell c is either empty or a union of Vertices v Edges e (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

Geometric boundary Geometric boundary of a cell c is either empty or a union of Vertices v Edges e Faces f (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

Geometric boundary Geometric boundary of a cell c is either empty or a union of Vertices v Edges e Faces f Solids s (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

Geometric boundary Geometric boundary of a cell c is either empty or a union of Vertices v Edges e Faces f Solids s ∂ v = ∅ (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

Geometric boundary Geometric boundary of a cell c is either empty or a union of Vertices v Edges e Faces f Solids s ∂ v = ∅ ∂ e = S 0 (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

Geometric boundary Geometric boundary of a cell c is either empty or a union of Vertices v Edges e Faces f Solids s ∂ v = ∅ ∂ e = S 0 ∂ f = S 1 (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

Geometric boundary Geometric boundary of a cell c is either empty or a union of Vertices v Edges e Faces f Solids s ∂ v = ∅ ∂ e = S 0 ∂ f = S 1 ∂ s = S 2 (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

Geometric boundary Geometric boundary of a cell c is either empty or a union of Vertices v Edges e Faces f Solids s ∂ v = ∅ ∂ e = S 0 ∂ f = S 1 ∂ s = S 2 ∂ ( ∂ c ) = ∅ (spheres have empty boundary) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

The boundary operator Define ∂ : C ( X ) → C ( X ) to be zero on vertices and linear on chains (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 26 / 37

The boundary operator Define ∂ : C ( X ) → C ( X ) to be zero on vertices and linear on chains I ∈ C ( I ) : ∂ I = 0 + 1 (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 26 / 37

The boundary operator Define ∂ : C ( X ) → C ( X ) to be zero on vertices and linear on chains I ∈ C ( I ) : ∂ I = 0 + 1 a + b ∈ C ( 8 ) : ∂ ( a + b ) = ∂ a + ∂ b = ( v + v ) + ( v + v ) = 0 (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 26 / 37

The boundary operator Define ∂ : C ( X ) → C ( X ) to be zero on vertices and linear on chains I ∈ C ( I ) : ∂ I = 0 + 1 a + b ∈ C ( 8 ) : ∂ ( a + b ) = ∂ a + ∂ b = ( v + v ) + ( v + v ) = 0 I × I ∈ C ( I × I ) : ∂ ( I × I ) = 0 × I + 1 × I + I × 0 + I × 1 = ( 0 + 1 ) × I + I × ( 0 + 1 ) = ∂ I × I + I × ∂ I (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 26 / 37

The boundary operator Define ∂ : C ( X ) → C ( X ) to be zero on vertices and linear on chains I ∈ C ( I ) : ∂ I = 0 + 1 a + b ∈ C ( 8 ) : ∂ ( a + b ) = ∂ a + ∂ b = ( v + v ) + ( v + v ) = 0 I × I ∈ C ( I × I ) : ∂ ( I × I ) = 0 × I + 1 × I + I × 0 + I × 1 = ( 0 + 1 ) × I + I × ( 0 + 1 ) = ∂ I × I + I × ∂ I ∂ is a derivation of Cartesian product! (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 26 / 37

Example Cells of M and T have empty boundaries (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 27 / 37

Example Cells of M and T have empty boundaries ∂ ≡ 0 on C ( M ) and C ( T ) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 27 / 37

Example Cells of M and T have empty boundaries ∂ ≡ 0 on C ( M ) and C ( T ) ∂ : C ( UN ) → C ( UN ) is defined ∂ v = ∂ a = ∂ b = ∂ s = ∂ t 1 = ∂ t 2 = 0 ∂ p = s ∂ q = s + t 1 + t 2 (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 27 / 37

Example Cells of M and T have empty boundaries ∂ ≡ 0 on C ( M ) and C ( T ) ∂ : C ( UN ) → C ( UN ) is defined ∂ v = ∂ a = ∂ b = ∂ s = ∂ t 1 = ∂ t 2 = 0 ∂ p = s ∂ q = s + t 1 + t 2 ∂ : C ( LN ) → C ( LN ) is defined ∂ a = ∂ b = ∂ s = ∂ t � 1 = ∂ t � ∂ v = 2 = 0 ∂ p = s ∂ q � s + t � 1 + t � = 2 (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 27 / 37

Cellular homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 28 / 37

Cellular homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ Cellular homology of X : H ( X ) = ker ∂ / Im ∂ (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 28 / 37

Cellular homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ Cellular homology of X : H ( X ) = ker ∂ / Im ∂ Elements are additive cosets [ c ] : = c + Im ∂ (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 28 / 37

Cellular homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ Cellular homology of X : H ( X ) = ker ∂ / Im ∂ Elements are additive cosets [ c ] : = c + Im ∂ Example: H ( M ) = C ( M ) and H ( T ) = C ( T ) ( ∂ ≡ 0) H ( M ) = C ( M ) ≈ C ( T ) = H ( T ) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 28 / 37

Cellular homology Example: H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } [ t 1 ] = t 1 + Im ∂ = t 1 + { 0 , s , t 1 + t 2 , s + t 1 + t 2 } (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 29 / 37

Cellular homology Example: H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } [ t 1 ] = t 1 + Im ∂ = t 1 + { 0 , s , t 1 + t 2 , s + t 1 + t 2 } Example: H ( LN ) = { [ v ] , [ a ] , [ b ] , [ t � 1 ] = [ t � 2 ] } (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 29 / 37

Cellular homology Example: H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } [ t 1 ] = t 1 + Im ∂ = t 1 + { 0 , s , t 1 + t 2 , s + t 1 + t 2 } Example: H ( LN ) = { [ v ] , [ a ] , [ b ] , [ t � 1 ] = [ t � 2 ] } H ( UN ) ≈ H ( LN ) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 29 / 37

Cellular homology Example: H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } [ t 1 ] = t 1 + Im ∂ = t 1 + { 0 , s , t 1 + t 2 , s + t 1 + t 2 } Example: H ( LN ) = { [ v ] , [ a ] , [ b ] , [ t � 1 ] = [ t � 2 ] } H ( UN ) ≈ H ( LN ) How does ∆ play out in homology? (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 29 / 37

Homology of Cartesian products Consider vector space A with basis { a 1 , . . . , a k } (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 30 / 37

Homology of Cartesian products Consider vector space A with basis { a 1 , . . . , a k } Tensor product A ⊗ A is the vector space with basis { a i ⊗ a j } 1 ≤ i ≤ k (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 30 / 37

Homology of Cartesian products Consider vector space A with basis { a 1 , . . . , a k } Tensor product A ⊗ A is the vector space with basis { a i ⊗ a j } 1 ≤ i ≤ k C ( X × X ) ≈ C ( X ) ⊗ C ( X ) via e × e � �→ e ⊗ e � (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 30 / 37

Homology of Cartesian products Consider vector space A with basis { a 1 , . . . , a k } Tensor product A ⊗ A is the vector space with basis { a i ⊗ a j } 1 ≤ i ≤ k C ( X × X ) ≈ C ( X ) ⊗ C ( X ) via e × e � �→ e ⊗ e � ∂ X × X + X × ∂ X = ( ∂ × Id + Id × ∂ ) ( X × X ) induces ∂ ⊗ Id + Id ⊗ ∂ : C ( X ) ⊗ C ( X ) → C ( X ) ⊗ C ( X ) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 30 / 37

Homology of Cartesian products Consider vector space A with basis { a 1 , . . . , a k } Tensor product A ⊗ A is the vector space with basis { a i ⊗ a j } 1 ≤ i ≤ k C ( X × X ) ≈ C ( X ) ⊗ C ( X ) via e × e � �→ e ⊗ e � ∂ X × X + X × ∂ X = ( ∂ × Id + Id × ∂ ) ( X × X ) induces ∂ ⊗ Id + Id ⊗ ∂ : C ( X ) ⊗ C ( X ) → C ( X ) ⊗ C ( X ) ( ∂ ⊗ Id + Id ⊗ ∂ ) 2 = ∂ 2 ⊗ Id 2 + 2 ∂ ⊗ ∂ + Id 2 ⊗ ∂ 2 = 0 (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 30 / 37

Homology of Cartesian products Consider vector space A with basis { a 1 , . . . , a k } Tensor product A ⊗ A is the vector space with basis { a i ⊗ a j } 1 ≤ i ≤ k C ( X × X ) ≈ C ( X ) ⊗ C ( X ) via e × e � �→ e ⊗ e � ∂ X × X + X × ∂ X = ( ∂ × Id + Id × ∂ ) ( X × X ) induces ∂ ⊗ Id + Id ⊗ ∂ : C ( X ) ⊗ C ( X ) → C ( X ) ⊗ C ( X ) ( ∂ ⊗ Id + Id ⊗ ∂ ) 2 = ∂ 2 ⊗ Id 2 + 2 ∂ ⊗ ∂ + Id 2 ⊗ ∂ 2 = 0 ∂ ⊗ Id + Id ⊗ ∂ is a boundary operator on C ( X ) ⊗ C ( X ) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 30 / 37

Homology of Cartesian products Consider vector space A with basis { a 1 , . . . , a k } Tensor product A ⊗ A is the vector space with basis { a i ⊗ a j } 1 ≤ i ≤ k C ( X × X ) ≈ C ( X ) ⊗ C ( X ) via e × e � �→ e ⊗ e � ∂ X × X + X × ∂ X = ( ∂ × Id + Id × ∂ ) ( X × X ) induces ∂ ⊗ Id + Id ⊗ ∂ : C ( X ) ⊗ C ( X ) → C ( X ) ⊗ C ( X ) ( ∂ ⊗ Id + Id ⊗ ∂ ) 2 = ∂ 2 ⊗ Id 2 + 2 ∂ ⊗ ∂ + Id 2 ⊗ ∂ 2 = 0 ∂ ⊗ Id + Id ⊗ ∂ is a boundary operator on C ( X ) ⊗ C ( X ) H ( X × X ) ≈ H ( X ) ⊗ H ( X ) (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 30 / 37

The induced diagonal on H(X) If ∆ : X → X × X is a diagonal approximation and c ∈ C ( X ) , define ∆ H : H ( X ) → H ( X ) ⊗ H ( X ) by ∆ H [ c ] : = [ ∆ c ] (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 31 / 37

The induced diagonal on H(X) If ∆ : X → X × X is a diagonal approximation and c ∈ C ( X ) , define ∆ H : H ( X ) → H ( X ) ⊗ H ( X ) by ∆ H [ c ] : = [ ∆ c ] Example: Every class of H ( M ) = C ( M ) is a singleton class ⇒ ∆ H [ M ] = [ ∆ M ] = [ v ⊗ M + M ⊗ v ] = v ⊗ M + M ⊗ v (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 31 / 37

The induced diagonal on H(X) If ∆ : X → X × X is a diagonal approximation and c ∈ C ( X ) , define ∆ H : H ( X ) → H ( X ) ⊗ H ( X ) by ∆ H [ c ] : = [ ∆ c ] Example: Every class of H ( M ) = C ( M ) is a singleton class ⇒ ∆ H [ M ] = [ ∆ M ] = [ v ⊗ M + M ⊗ v ] = v ⊗ M + M ⊗ v Example: Every class of H ( T ) = C ( T ) is a singleton class ⇒ ∆ H [ T ] = [ ∆ T ] = v ⊗ T + T ⊗ v + a ⊗ b + b ⊗ a (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 31 / 37

Homotopy invariance Fact: Homotopy is an equivalence relation on { cont f : X → X × X } (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 32 / 37

Homotopy invariance Fact: Homotopy is an equivalence relation on { cont f : X → X × X } Key fact: Homotopic maps induce identical maps on homology (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 32 / 37

Homotopy invariance Fact: Homotopy is an equivalence relation on { cont f : X → X × X } Key fact: Homotopic maps induce identical maps on homology Diagonal approximations induce identical maps on homology (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 32 / 37

Homotopy invariance Fact: Homotopy is an equivalence relation on { cont f : X → X × X } Key fact: Homotopic maps induce identical maps on homology Diagonal approximations induce identical maps on homology Thus ∆ H [ M ] � = ∆ H [ T ] implies M � T (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 32 / 37

Homotopy invariance Fact: Homotopy is an equivalence relation on { cont f : X → X × X } Key fact: Homotopic maps induce identical maps on homology Diagonal approximations induce identical maps on homology Thus ∆ H [ M ] � = ∆ H [ T ] implies M � T But we already knew this... so what’s the big deal? (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 32 / 37

Homotopy invariance ∆ H [ t 1 ] = [ ∆ t 1 ] = [ v ⊗ t 1 + t 1 ⊗ v ] = [ v ] ⊗ [ t 1 ] + [ t 1 ] ⊗ [ v ] (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 33 / 37

Homotopy invariance ∆ H [ t 1 ] = [ ∆ t 1 ] = [ v ⊗ t 1 + t 1 ⊗ v ] = [ v ] ⊗ [ t 1 ] + [ t 1 ] ⊗ [ v ] ∆ H [ t � 1 ] = [ ∆ t � 1 ] = [ v ⊗ t � 1 + t � 1 ⊗ v + a ⊗ b + b ⊗ a ] = [ v ] ⊗ [ t � 1 ] + [ t � 1 ] ⊗ [ v ] + [ a ] ⊗ [ b ] + [ b ] ⊗ [ a ] (MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 33 / 37