Detecting Linkage in an n -Component Brunnian Link (work in - PowerPoint PPT Presentation

The Geometric Diagonal Geometric diagonal ∆ X : X → X × X is defined x �→ ( x , x ) A homeomorphism h : X → Y respects diagonals ∆ Y h = ( h × h ) ∆ X Objective: Compute the obstruction to a homeomorphism h : UN → LN Strategy: Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 14 / 28

The Geometric Diagonal Geometric diagonal ∆ X : X → X × X is defined x �→ ( x , x ) A homeomorphism h : X → Y respects diagonals ∆ Y h = ( h × h ) ∆ X Objective: Compute the obstruction to a homeomorphism h : UN → LN Strategy: Assume there is a homeomorphism h Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 14 / 28

The Geometric Diagonal Geometric diagonal ∆ X : X → X × X is defined x �→ ( x , x ) A homeomorphism h : X → Y respects diagonals ∆ Y h = ( h × h ) ∆ X Objective: Compute the obstruction to a homeomorphism h : UN → LN Strategy: Assume there is a homeomorphism h Show that h fails to respect diagonals Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 14 / 28

The Geometric Diagonal Problem: Im ∆ X is typically not a subcomplex of X × X Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 15 / 28

The Geometric Diagonal Problem: Im ∆ X is typically not a subcomplex of X × X Example: Im ∆ I is not a subcomplex of I × I : Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 15 / 28

Diagonal Approximations A map ∆ : X → X × X is a diagonal approximation if Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 16 / 28

Diagonal Approximations A map ∆ : X → X × X is a diagonal approximation if ∆ is homotopic to ∆ X Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 16 / 28

Diagonal Approximations A map ∆ : X → X × X is a diagonal approximation if ∆ is homotopic to ∆ X ∆ ( e n ) is a subcomplex of e n × e n for every n -cell e n ⊆ X Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 16 / 28

Diagonal Approximations A map ∆ : X → X × X is a diagonal approximation if ∆ is homotopic to ∆ X ∆ ( e n ) is a subcomplex of e n × e n for every n -cell e n ⊆ X Geometric boundary ∂ : X → X is a coderivation of ∆ ∆ ∂ = ( ∂ × Id + Id × ∂ ) ∆ Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 16 / 28

Diagonal Approximations A map ∆ : X → X × X is a diagonal approximation if ∆ is homotopic to ∆ X ∆ ( e n ) is a subcomplex of e n × e n for every n -cell e n ⊆ X Geometric boundary ∂ : X → X is a coderivation of ∆ ∆ ∂ = ( ∂ × Id + Id × ∂ ) ∆ Cellular Approximation Theorem There is a diagonal approximation ∆ : X → X × X Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 16 / 28

Diagonal Approximations Preserve Cellular structure: ∆ ( e n ) ⊆ e n × e n Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 17 / 28

Diagonal Approximations Preserve Cellular structure: ∆ ( e n ) ⊆ e n × e n Dimension: dim ∆ ( e n ) = dim e n Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 17 / 28

Diagonal Approximations Preserve Cellular structure: ∆ ( e n ) ⊆ e n × e n Dimension: dim ∆ ( e n ) = dim e n Cartesian products: ∆ ( X × Y ) = ∆ ( X ) × ∆ ( Y ) Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 17 / 28

Diagonal Approximations Preserve Cellular structure: ∆ ( e n ) ⊆ e n × e n Dimension: dim ∆ ( e n ) = dim e n Cartesian products: ∆ ( X × Y ) = ∆ ( X ) × ∆ ( Y ) Wedge products: ∆ ( X ∨ Y ) = ∆ ( X ) ∨ ∆ ( Y ) Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 17 / 28

Dan Kravatz’s Diagonal Approximation on a Polygon Given n -gon G , arbitrarily choose vertices v and v � (possibly equal) Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 18 / 28

Dan Kravatz’s Diagonal Approximation on a Polygon Given n -gon G , arbitrarily choose vertices v and v � (possibly equal) Edges { e 1 , . . . , e k } and { e k + 1 , . . . , e n } form paths from v to v � (one path { e 1 , . . . , e n } if v = v � ) Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 18 / 28

Dan Kravatz’s Diagonal Approximation on a Polygon Given n -gon G , arbitrarily choose vertices v and v � (possibly equal) Edges { e 1 , . . . , e k } and { e k + 1 , . . . , e n } form paths from v to v � (one path { e 1 , . . . , e n } if v = v � ) Theorem (Kravatz 2008): There is a diagonal approximation ∆ G = v × G + G × v � k ∑ + ( e 1 + · · · + e i − 1 ) × e i i = 2 n ∑ + ( e k + 1 + · · · + e j − 1 ) × e j j = k + 2 Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 18 / 28

Example Think of the pinched sphere t 1 ⊂ ∂ ( UN ) as a 2-gon with vertices identified first, then edges identified ∆ t 1 = v × t 1 + t 1 × v ∆ descends to quotients when edge-paths are consistent with identifications Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 19 / 28

Example Think of the torus t � 1 ⊂ ∂ ( LN ) as a square with horizontal edges a identified and vertical edges b identified ∆ t � 1 = v × t � 1 + t � 1 × v + a × b + b × a Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 20 / 28

Cellular Chains C ( X ) denotes the Z 2 -vector space with basis { cells of X } Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 21 / 28

Cellular Chains C ( X ) denotes the Z 2 -vector space with basis { cells of X } Elements are formal sums called cellular chains of X Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 21 / 28

Cellular Chains C ( X ) denotes the Z 2 -vector space with basis { cells of X } Elements are formal sums called cellular chains of X Examples: Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 21 / 28

Cellular Chains C ( X ) denotes the Z 2 -vector space with basis { cells of X } Elements are formal sums called cellular chains of X Examples: C ( UN ) has basis { v , a , b , s , t 1 , t 2 , p , q } Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 21 / 28

Cellular Chains C ( X ) denotes the Z 2 -vector space with basis { cells of X } Elements are formal sums called cellular chains of X Examples: 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 � } Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 21 / 28

Cellular Chains C ( X ) denotes the Z 2 -vector space with basis { cells of X } Elements are formal sums called cellular chains of X Examples: 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 ) Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 21 / 28

The Boundary Operator Geometric boundary of an n -cell e n is S n − 1 Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 22 / 28

The Boundary Operator Geometric boundary of an n -cell e n is S n − 1 ∂ v = ∅ ; ∂ e = S 0 ; ∂ f = S 1 ; ∂ s = S 2 Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 22 / 28

The Boundary Operator Geometric boundary of an n -cell e n is S n − 1 ∂ v = ∅ ; ∂ e = S 0 ; ∂ f = S 1 ; ∂ s = S 2 ∂ ( ∂ e n ) = ∂ S n − 1 = ∅ Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 22 / 28

The Boundary Operator Geometric boundary of an n -cell e n is S n − 1 ∂ v = ∅ ; ∂ e = S 0 ; ∂ f = S 1 ; ∂ s = S 2 ∂ ( ∂ e n ) = ∂ S n − 1 = ∅ Boundary operator ∂ : C ( X ) → C ( X ) Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 22 / 28

The Boundary Operator Geometric boundary of an n -cell e n is S n − 1 ∂ v = ∅ ; ∂ e = S 0 ; ∂ f = S 1 ; ∂ s = S 2 ∂ ( ∂ e n ) = ∂ S n − 1 = ∅ Boundary operator ∂ : C ( X ) → C ( X ) Induced by the geometric boundary Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 22 / 28

The Boundary Operator Geometric boundary of an n -cell e n is S n − 1 ∂ v = ∅ ; ∂ e = S 0 ; ∂ f = S 1 ; ∂ s = S 2 ∂ ( ∂ e n ) = ∂ S n − 1 = ∅ Boundary operator ∂ : C ( X ) → C ( X ) Induced by the geometric boundary Zero on vertices Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 22 / 28

The Boundary Operator Geometric boundary of an n -cell e n is S n − 1 ∂ v = ∅ ; ∂ e = S 0 ; ∂ f = S 1 ; ∂ s = S 2 ∂ ( ∂ e n ) = ∂ S n − 1 = ∅ Boundary operator ∂ : C ( X ) → C ( X ) Induced by the geometric boundary Zero on vertices Linear on chains Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 22 / 28

The Boundary Operator Geometric boundary of an n -cell e n is S n − 1 ∂ v = ∅ ; ∂ e = S 0 ; ∂ f = S 1 ; ∂ s = S 2 ∂ ( ∂ e n ) = ∂ S n − 1 = ∅ Boundary operator ∂ : C ( X ) → C ( X ) Induced by the geometric boundary Zero on vertices Linear on chains A derivation of the Cartesian product ∂ ( a × b ) = ∂ a × b + a × ∂ b Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 22 / 28

Examples ∂ : C ( UN ) → C ( UN ) is defined ∂ v = ∂ a = ∂ b = ∂ s = ∂ t 1 = ∂ t 2 = 0 ∂ p = s ∂ q = s + t 1 + t 2 Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 23 / 28

Examples ∂ : 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 ∂ v = ∂ a = ∂ b = ∂ s = ∂ t � 1 = ∂ t � 2 = 0 ∂ p = s ∂ q � = s + t � 1 + t � 2 Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 23 / 28

Cellular Homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 24 / 28

Cellular Homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ H ( X ) : = ker ∂ / Im ∂ is the cellular homology of X Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 24 / 28

Cellular Homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ H ( X ) : = ker ∂ / Im ∂ is the cellular homology of X Elements of H ( X ) are cosets [ c ] : = c + Im ∂ Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 24 / 28

Cellular Homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ H ( X ) : = ker ∂ / Im ∂ is the cellular homology of X Elements of H ( X ) are cosets [ c ] : = c + Im ∂ Examples Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 24 / 28

Cellular Homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ H ( X ) : = ker ∂ / Im ∂ is the cellular homology of X Elements of H ( X ) are cosets [ c ] : = c + Im ∂ Examples H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 24 / 28

Cellular Homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ H ( X ) : = ker ∂ / Im ∂ is the cellular homology of X Elements of H ( X ) are cosets [ c ] : = c + Im ∂ Examples H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } H ( LN ) = { [ v ] , [ a ] , [ b ] , [ t � 1 ] = [ t � 2 ] } Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 24 / 28

Cellular Homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ H ( X ) : = ker ∂ / Im ∂ is the cellular homology of X Elements of H ( X ) are cosets [ c ] : = c + Im ∂ Examples H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } H ( LN ) = { [ v ] , [ a ] , [ b ] , [ t � 1 ] = [ t � 2 ] } Note that H ( UN ) ≈ H ( LN ) Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 24 / 28

Cellular Homology ∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂ H ( X ) : = ker ∂ / Im ∂ is the cellular homology of X Elements of H ( X ) are cosets [ c ] : = c + Im ∂ Examples H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } H ( LN ) = { [ v ] , [ a ] , [ b ] , [ t � 1 ] = [ t � 2 ] } Note that H ( UN ) ≈ H ( LN ) How do diagonal approximations on UN and LN lift to homology? Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 24 / 28

Key Facts Homotopic maps of spaces induce the same map on their homologies Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 25 / 28

Key Facts Homotopic maps of spaces induce the same map on their homologies Every diagonal approximation ∆ : X → X × X induces the same map ∆ 2 : H ( X ) → H ( X × X ) Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 25 / 28

Key Facts Homotopic maps of spaces induce the same map on their homologies Every diagonal approximation ∆ : X → X × X induces the same map ∆ 2 : H ( X ) → H ( X × X ) A homeomorphism h : X → Y induces maps h ∗ : H ( X ) → H ( Y ) and ( h × h ) ∗ : H ( X × X ) → H ( Y × Y ) such that ∆ 2 h ∗ = ( h × h ) ∗ ∆ 2 Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 25 / 28

Key Facts Homotopic maps of spaces induce the same map on their homologies Every diagonal approximation ∆ : X → X × X induces the same map ∆ 2 : H ( X ) → H ( X × X ) A homeomorphism h : X → Y induces maps h ∗ : H ( X ) → H ( Y ) and ( h × h ) ∗ : H ( X × X ) → H ( Y × Y ) such that ∆ 2 h ∗ = ( h × h ) ∗ ∆ 2 If h : UN → LN is a homeomorphism, inequality is a contradiction Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 25 / 28

Homology of Cartesian Products If vector space A has basis { a 1 , . . . , a k } , the tensor product vector space A ⊗ A has basis { a i ⊗ a j } 1 ≤ i , j ≤ k Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 26 / 28

Homology of Cartesian Products If vector space A has basis { a 1 , . . . , a k } , the tensor product vector space A ⊗ A has basis { a i ⊗ a j } 1 ≤ i , j ≤ k C ( X × X ) ≈ C ( X ) ⊗ C ( X ) via e × e � �→ e ⊗ e � Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 26 / 28

Homology of Cartesian Products If vector space A has basis { a 1 , . . . , a k } , the tensor product vector space A ⊗ A has basis { a i ⊗ a j } 1 ≤ i , j ≤ k C ( X × X ) ≈ C ( X ) ⊗ C ( X ) via e × e � �→ e ⊗ e � The boundary map ∂ × Id + Id × ∂ : X × X → X × X induces the boundary operator ∂ ⊗ Id + Id ⊗ ∂ : C ( X ) ⊗ C ( X ) → C ( X ) ⊗ C ( X ) Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 26 / 28

Homology of Cartesian Products If vector space A has basis { a 1 , . . . , a k } , the tensor product vector space A ⊗ A has basis { a i ⊗ a j } 1 ≤ i , j ≤ k C ( X × X ) ≈ C ( X ) ⊗ C ( X ) via e × e � �→ e ⊗ e � The boundary map ∂ × Id + Id × ∂ : X × X → X × X induces the boundary operator ∂ ⊗ Id + Id ⊗ ∂ : C ( X ) ⊗ C ( X ) → C ( X ) ⊗ C ( X ) Since Z 2 is a field, torsion vanishes and H ( X × X ) ≈ H ( X ) ⊗ H ( X ) Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 26 / 28

Induced Diagonal on H(X) A diagonal approximation ∆ : X → X × X induces a coproduct ∆ 2 : H ( X ) → H ( X ) ⊗ H ( X ) defined by ∆ 2 [ c ] : = [ ∆ c ] Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 27 / 28

Induced Diagonal on H(X) A diagonal approximation ∆ : X → X × X induces a coproduct ∆ 2 : H ( X ) → H ( X ) ⊗ H ( X ) defined by ∆ 2 [ c ] : = [ ∆ c ] A class [ c ] of positive dimension is primitive if ∆ 2 [ c ] = [ v ] ⊗ [ c ] + [ c ] ⊗ [ v ] Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 27 / 28

Induced Diagonal on H(X) A diagonal approximation ∆ : X → X × X induces a coproduct ∆ 2 : H ( X ) → H ( X ) ⊗ H ( X ) defined by ∆ 2 [ c ] : = [ ∆ c ] A class [ c ] of positive dimension is primitive if ∆ 2 [ c ] = [ v ] ⊗ [ c ] + [ c ] ⊗ [ v ] Examples ∆ 2 [ t 1 ] = [ ∆ t 1 ] = [ v ] ⊗ [ t 1 ] + [ t 1 ] ⊗ [ v ] � � = � � = [ v ] ⊗ � � + � � ⊗ [ v ] + [ a ] ⊗ [ b ] + [ b ] ⊗ [ a ] t � ∆ t � t � t � ∆ 2 1 1 1 1 Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 27 / 28

Non-Primitivity Detects Linkage If h : UN → LN is a homeomorphism, ( h ∗ ⊗ h ∗ ) ∆ 2 = ∆ 2 h ∗ Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 28 / 28

Non-Primitivity Detects Linkage If h : UN → LN is a homeomorphism, ( h ∗ ⊗ h ∗ ) ∆ 2 = ∆ 2 h ∗ But h ∗ [ t 1 ] = [ t � 1 ] implies ( h ∗ ⊗ h ∗ ) ∆ 2 [ t 1 ] = ( h ∗ ⊗ h ∗ ) ([ v ] ⊗ [ t 1 ] + [ t 1 ] ⊗ [ v ]) � � + � � ⊗ [ v ] t � t � = [ v ] ⊗ 1 1 � � + � � ⊗ [ v ] + [ a ] ⊗ [ b ] + [ b ] ⊗ [ a ] t � t � � = [ v ] ⊗ 1 1 � � = ∆ 2 h ∗ [ t 1 ] , t � = ∆ 2 1 which is a contradiction Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 28 / 28

Non-Primitivity Detects Linkage If h : UN → LN is a homeomorphism, ( h ∗ ⊗ h ∗ ) ∆ 2 = ∆ 2 h ∗ But h ∗ [ t 1 ] = [ t � 1 ] implies ( h ∗ ⊗ h ∗ ) ∆ 2 [ t 1 ] = ( h ∗ ⊗ h ∗ ) ([ v ] ⊗ [ t 1 ] + [ t 1 ] ⊗ [ v ]) � � + � � ⊗ [ v ] t � t � = [ v ] ⊗ 1 1 � � + � � ⊗ [ v ] + [ a ] ⊗ [ b ] + [ b ] ⊗ [ a ] t � t � � = [ v ] ⊗ 1 1 � � = ∆ 2 h ∗ [ t 1 ] , t � = ∆ 2 1 which is a contradiction The non-primitive coproduct has detected the Hopf Link! Tetrahedral Geometry/Topology Seminar ( Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ) December 4, 2015 28 / 28

Recap Homology alone cannot distinguish links from unlinks. Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 1 / 12

Recap Homology alone cannot distinguish links from unlinks. For example, H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } � = t ′ t ′ � [ v ] , [ a ] , [ b ] , � � �� and H ( LN ) = , 1 2 so H ( UN ) ≈ H ( LN ) . Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 1 / 12

Recap Homology alone cannot distinguish links from unlinks. For example, H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } � = t ′ t ′ � [ v ] , [ a ] , [ b ] , � � �� and H ( LN ) = , 1 2 so H ( UN ) ≈ H ( LN ) . But perhaps homology with additional structure derived from diagonal approximations can distinguish between links. Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 1 / 12

Recap Homology alone cannot distinguish links from unlinks. For example, H ( UN ) = { [ v ] , [ a ] , [ b ] , [ t 1 ] = [ t 2 ] } � = t ′ t ′ � [ v ] , [ a ] , [ b ] , � � �� and H ( LN ) = , 1 2 so H ( UN ) ≈ H ( LN ) . But perhaps homology with additional structure derived from diagonal approximations can distinguish between links. For example, the coproducts, ∆ 2 , induced by diagonal approximations are different for UN and LN . ∆ 2 [ t 1 ] = [ v ] ⊗ [ t 1 ] + [ t 1 ] ⊗ [ v ] is primitive. � + � ⊗ [ v ] + [ a ] ⊗ [ b ] + [ b ] ⊗ [ a ] t ′ t ′ t ′ � � = [ v ] ⊗ � � ∆ 2 1 1 1 is not primitive. Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 1 / 12

Definition Definition: A nontrivial link is called Brunnian if it has the following property: Removing any component results in an unlink. Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 2 / 12

Definition Definition: A nontrivial link is called Brunnian if it has the following property: Removing any component results in an unlink. (a non-standard) Example: The Hopf link. Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 2 / 12

Definition Definition: A nontrivial link is called Brunnian if it has the following property: Removing any component results in an unlink. (a non-standard) Example: The Hopf link. (the most standard) Example: The Borromean rings. Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 2 / 12

Definition Definition: A nontrivial link is called Brunnian if it has the following property: Removing any component results in an unlink. (a non-standard) Example: The Hopf link. (the most standard) Example: The Borromean rings. Let BR n denote the complement in S 3 of a Brunnian Convention: link with n components where n > 3. Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 2 / 12

Conjecture Conjecture: A diagonal approximation ∆ on C ( BR n ) induces a primitive diagonal ∆ 2 : H ( BR n ) → H ( BR n ) ⊗ H ( BR n ) , Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 3 / 12

Conjecture Conjecture: A diagonal approximation ∆ on C ( BR n ) induces a primitive diagonal ∆ 2 : H ( BR n ) → H ( BR n ) ⊗ H ( BR n ) , trivial k-ary operations ∆ k : H ( BR n ) → H ( BR n ) ⊗ k for 3 ≤ k < n, and Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 3 / 12

Conjecture Conjecture: A diagonal approximation ∆ on C ( BR n ) induces a primitive diagonal ∆ 2 : H ( BR n ) → H ( BR n ) ⊗ H ( BR n ) , trivial k-ary operations ∆ k : H ( BR n ) → H ( BR n ) ⊗ k for 3 ≤ k < n, and a non-trivial n-ary operation ∆ n : H ( BR ) → H ( BR ) ⊗ n . Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 3 / 12

Work to do Describe an infinite family of Brunnian links, including cell decompositions and diagonal approximations. Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 4 / 12

Work to do Describe an infinite family of Brunnian links, including cell decompositions and diagonal approximations. (computationally) Transfer the differential graded coalgebra structure on the chains to homology. Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 4 / 12

Work to do Describe an infinite family of Brunnian links, including cell decompositions and diagonal approximations. (computationally) Transfer the differential graded coalgebra structure on the chains to homology. (hopefully) Observe the conjectured results. Tetrahedral Geometry/Topology Seminar Detecting Brunnian Linkage ( Tetrahedral Geometry/Topology Seminar December 4, 2015 ) 4 / 12