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

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

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Homology? Its Mickey Mouse! So whats the big deal? Ron Umble, speaker Millersville Univ of Pennsylvania MU/F&M Mathematics Colloquium April 4, 2013 (MU/F&M Mathematics Colloquium ) Homology? Its Mickey Mouse! April 4, 2013

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Homology? It’s Mickey Mouse!

So what’s the big deal? Ron Umble, speaker Millersville Univ of Pennsylvania

MU/F&M Mathematics Colloquium

April 4, 2013

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 1 / 37

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Cellular decompositions

X denotes a network, surface, solid or union thereof embedded in R3 or S3

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 2 / 37

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Cellular decompositions

A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells)

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 3 / 37

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Cellular decompositions

A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells)

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 3 / 37

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Cellular decompositions

A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) closed disks (faces or 2-cells)

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 3 / 37

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Cellular decompositions

A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) closed disks (faces or 2-cells) closed balls (solids or 3-cells) which have been glued together in such a way that the

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 3 / 37

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SLIDE 7

Cellular decompositions

A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) closed disks (faces or 2-cells) closed balls (solids or 3-cells) which have been glued together in such a way that the non-empty boundary of a k-cell is a union of (k − 1)-cells

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 3 / 37

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Cellular decompositions

A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) closed disks (faces or 2-cells) closed balls (solids or 3-cells) which have been glued together in such a way that the non-empty boundary of a k-cell is a union of (k − 1)-cells non-empty intersection of cells is a cell

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 3 / 37

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Cellular decompositions

A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) closed disks (faces or 2-cells) closed balls (solids or 3-cells) which have been glued together in such a way that the non-empty boundary of a k-cell is a union of (k − 1)-cells non-empty intersection of cells is a cell union of all cells is X

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 3 / 37

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Example

Unit interval I = [0, 1] Vertices: {0, 1} Edge: {I}

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 4 / 37

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Example

Unit square I2 = I × I Product cells: {0, 1, I} × {0, 1, I} Vertices: {00, 01, 10, 11} Edges: {0I, 1I, I0, I1} Face:

  • I2

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 5 / 37

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Example

Circle S1 — endpoints of I identified Vertex: {v} Edge: {a}

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 6 / 37

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SLIDE 13

Examples

Figure 8 = S1 ∨ S1 Vertex: {v} Edges: {a, b}

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 7 / 37

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Example

Sphere S2 — boundary of a closed disk is identified (draw-string bag) Vertex: {v} Edges: ∅ Face: {S2}

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 8 / 37

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Example

Mickey Mouse (with pierced ears!) M = S1 ∨ S1 ∨ S2 Vertex: {v} Edges: {a, b} Face:

  • S2

− → − →

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 9 / 37

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Example

Torus T = S1 × S1 Product cells: {v,a} × {v,b} Vertex: {v: = v × v} Edges: {a := a × v, b := v × b} Face: {T := a × b}

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 10 / 37

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Example

The complement UN of tubular neighborhoods of two unlinked loops in the lower hemisphere of S3; ∂ (UN) is the wedge of two pinched spheres; these are wedged with the equitorial 2-sphere in S3 Vertices: {v} Edges: {a, b} Faces: {s, t1, t2} Solids: {p, q} (UN = p ∪ q) The boundary of UN in S3

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 11 / 37

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Example

The complement LN of tubular neighborhoods of two linked loops in the lower hemishpere of S3; ∂ (LN) is two linked tori sharing a single vertex and two edges; these are wedged with the equitorial 2-sphere in S3 Vertex: {v} Edges: {a, b} Faces: {s, t

1, t 2}

Solids: {p, q} (LN = p ∪ q) The boundary of LN in S3

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 12 / 37

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Homeomorphisms

X and Y are homeomorphic if X continuously deforms into Y ∃ a bijective bicontinuous h : X → Y , called a homeomorphism

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 13 / 37

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Homeomorphisms

X and Y are homeomorphic if X continuously deforms into Y ∃ a bijective bicontinuous h : X → Y , called a homeomorphism A square and a circle are homeomorphic

h

− →

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 13 / 37

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Homeomorphisms

X and Y are homeomorphic if X continuously deforms into Y ∃ a bijective bicontinuous h : X → Y , called a homeomorphism A square and a circle are homeomorphic

h

− → The surfaces of a doughnut and coffee cup are homeomorphic Animation of morphing coffee mug

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 13 / 37

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Homeomorphisms

Fact: Homeomorphisms preserve number of connected components If ∀ bijections h : X → Y , ∃ x ∈ X s.t. X − {p} and Y − {h (p)} have different numbers of connected components, then X Y

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 14 / 37

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Homeomorphisms

Fact: Homeomorphisms preserve number of connected components If ∀ bijections h : X → Y , ∃ x ∈ X s.t. X − {p} and Y − {h (p)} have different numbers of connected components, then X Y If h : 8 → S1 is a bijection and p ∈ 8 is the double point, then 8 − {p} has two connected components and S1 − {h (p)} has one

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 14 / 37

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Homeomorphisms

Fact: Homeomorphisms preserve number of connected components If ∀ bijections h : X → Y , ∃ x ∈ X s.t. X − {p} and Y − {h (p)} have different numbers of connected components, then X Y If h : 8 → S1 is a bijection and p ∈ 8 is the double point, then 8 − {p} has two connected components and S1 − {h (p)} has one If h : M → T is a bijection, then M − {v} has three connected components and T − {h (v)} has one

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 14 / 37

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Homeomorphisms

Fact: Homeomorphisms preserve number of connected components If ∀ bijections h : X → Y , ∃ x ∈ X s.t. X − {p} and Y − {h (p)} have different numbers of connected components, then X Y If h : 8 → S1 is a bijection and p ∈ 8 is the double point, then 8 − {p} has two connected components and S1 − {h (p)} has one If h : M → T is a bijection, then M − {v} has three connected components and T − {h (v)} has one UN − {p} and LN − {q} are connected for all p ∈ UN and q ∈ LN... Hum...

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 14 / 37

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The geometric diagonal

Goal: Determine whether or not UL and LN are homeomorphic

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 15 / 37

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The geometric diagonal

Goal: Determine whether or not UL and LN are homeomorphic Geometric diagonal of X is the image of the map ∆G (x) = (x, x)

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 15 / 37

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The geometric diagonal

Goal: Determine whether or not UL and LN are homeomorphic Geometric diagonal of X is the image of the map ∆G (x) = (x, x) ∆G (X) is typically not a cell of X × X

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 15 / 37

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The geometric diagonal

Goal: Determine whether or not UL and LN are homeomorphic Geometric diagonal of X is the image of the map ∆G (x) = (x, x) ∆G (X) is typically not a cell of X × X Example: ∆G (I) is not a cell of I × I :

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 15 / 37

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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 f (t) =    (0, 2t) t ∈

  • 0, 1

2

  • (2t, 1)

t ∈ 1

2, 1

  • and g (t) =

   (2t, 0) t ∈

  • 0, 1

2

  • (1, 2t)

t ∈ 1

2, 1

  • (MU/F&M Mathematics Colloquium )

Homology? It’s Mickey Mouse! April 4, 2013 16 / 37

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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 f (t) =    (0, 2t) t ∈

  • 0, 1

2

  • (2t, 1)

t ∈ 1

2, 1

  • and g (t) =

   (2t, 0) t ∈

  • 0, 1

2

  • (1, 2t)

t ∈ 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

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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 f (t) =    (0, 2t) t ∈

  • 0, 1

2

  • (2t, 1)

t ∈ 1

2, 1

  • and g (t) =

   (2t, 0) t ∈

  • 0, 1

2

  • (1, 2t)

t ∈ 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

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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

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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

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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

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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

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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 ) Example: ∆

  • I2 = ∆ (I) × ∆ (I)

= (0 × I + I × 1) × (0 × I + I × 1) = 00 × I2 + 0I × I1 + I0 × 1I + I2 × 11

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 17 / 37

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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

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Dan Kravatz’s diagonal approximation on a polygon

Given n-gon G, n ≥ 1, choose arbitrary vertices v and v Edges {e1, . . . , ek} and {ek+1, . . . , en} form paths from v to v

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 18 / 37

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Dan Kravatz’s diagonal approximation on a polygon

Given n-gon G, n ≥ 1, choose arbitrary vertices v and v Edges {e1, . . . , ek} and {ek+1, . . . , en} form paths from v to v Theorem (Kravatz 2008): There is a diagonal approximation ∆G =

k

∑

i=2

(e1 + · · · + ei−1) × ei +

n

∑

j=k+2

(ek+1 + · · · + ej−1) × ej +v × G + G × v

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 18 / 37

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Example

Think of the sphere S2 as a 1-gon with its edge identified (v = v ) ∆S2 = v × S2 + S2 × v

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 19 / 37

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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

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Example

Think of Mickey Mouse M as the wedge of a 1-gon S2 with intervals a and b; the edge of S2 and the endpoints of a and b are identified ∆M = ∆

  • S2 ∨ S1 ∨ S1 = ∆S2 + ∆S1 + ∆S1

=

  • v × S2 + S2 × v

+ (v × a + a × v) + (v × b + b × v) = v ×

  • S2 + a + b

+

  • S2 + a + b

× v = v × M + M × v

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 21 / 37

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SLIDE 44

Example

Think of the pinched sphere t1 in ∂ (UN) as a 2-gon with vertices identified first, then edges identified ∆t1 = v × t1 + t1 × v

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 22 / 37

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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 ∆t1 = ∆t

1 UN LN

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 23 / 37

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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 ∆t1 = ∆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

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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 ∆t1 = ∆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

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Cellular chains of X

Elements of Z2-vector space C (X) with basis {cells of X}

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

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Cellular chains of X

Elements of Z2-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

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Cellular chains of X

Elements of Z2-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

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SLIDE 51

Cellular chains of X

Elements of Z2-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}

C (M) has basis

  • v, a, b, S2

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

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SLIDE 52

Cellular chains of X

Elements of Z2-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}

C (M) has basis

  • v, a, b, S2

C (T) has basis {v, a, b, T}

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

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SLIDE 53

Cellular chains of X

Elements of Z2-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}

C (M) has basis

  • v, a, b, S2

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

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SLIDE 54

Cellular chains of X

Elements of Z2-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}

C (M) has basis

  • v, a, b, S2

C (T) has basis {v, a, b, T}

Note that C (T) ≈ C (M)

C (UN) has basis {v, a, b, s, t1, t2, p, q}

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 24 / 37

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SLIDE 55

Cellular chains of X

Elements of Z2-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}

C (M) has basis

  • v, a, b, S2

C (T) has basis {v, a, b, T}

Note that C (T) ≈ C (M)

C (UN) has basis {v, a, b, s, t1, t2, 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

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SLIDE 56

Cellular chains of X

Elements of Z2-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}

C (M) has basis

  • v, a, b, S2

C (T) has basis {v, a, b, T}

Note that C (T) ≈ C (M)

C (UN) has basis {v, a, b, s, t1, t2, 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

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SLIDE 57

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

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SLIDE 58

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

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SLIDE 59

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

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SLIDE 60

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

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SLIDE 61

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

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SLIDE 62

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

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SLIDE 63

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 = S0

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

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SLIDE 64

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 = S0 ∂f = S1

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

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SLIDE 65

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 = S0 ∂f = S1 ∂s = S2

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

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SLIDE 66

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 = S0 ∂f = S1 ∂s = S2 ∂ (∂c) = ∅ (spheres have empty boundary)

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 25 / 37

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SLIDE 67

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

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SLIDE 68

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

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SLIDE 69

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

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SLIDE 70

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

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SLIDE 71

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

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SLIDE 72

Example

Cells of M and T have empty boundaries

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 27 / 37

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SLIDE 73

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

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SLIDE 74

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 = ∂t1 = ∂t2 = 0 ∂p = s ∂q = s + t1 + t2

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 27 / 37

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SLIDE 75

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 = ∂t1 = ∂t2 = 0 ∂p = s ∂q = s + t1 + t2 ∂ : C (LN) → C (LN) 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

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SLIDE 76

Cellular homology

∂ ◦ ∂ = 0 implies Im ∂ ⊆ ker ∂

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 28 / 37

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SLIDE 77

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

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SLIDE 78

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

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SLIDE 79

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

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SLIDE 80

Cellular homology

Example: H (UN) = {[v] , [a] , [b] , [t1] = [t2]} [t1] = t1 + Im ∂ = t1 + {0, s, t1 + t2, s + t1 + t2}

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 29 / 37

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SLIDE 81

Cellular homology

Example: H (UN) = {[v] , [a] , [b] , [t1] = [t2]} [t1] = t1 + Im ∂ = t1 + {0, s, t1 + t2, s + t1 + t2} 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

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SLIDE 82

Cellular homology

Example: H (UN) = {[v] , [a] , [b] , [t1] = [t2]} [t1] = t1 + Im ∂ = t1 + {0, s, t1 + t2, s + t1 + t2} 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

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SLIDE 83

Cellular homology

Example: H (UN) = {[v] , [a] , [b] , [t1] = [t2]} [t1] = t1 + Im ∂ = t1 + {0, s, t1 + t2, s + t1 + t2} 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

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SLIDE 84

Homology of Cartesian products

Consider vector space A with basis {a1, . . . , ak}

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 30 / 37

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SLIDE 85

Homology of Cartesian products

Consider vector space A with basis {a1, . . . , ak} Tensor product A ⊗ A is the vector space with basis {ai ⊗ aj}1≤i≤k

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SLIDE 86

Homology of Cartesian products

Consider vector space A with basis {a1, . . . , ak} Tensor product A ⊗ A is the vector space with basis {ai ⊗ aj}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

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SLIDE 87

Homology of Cartesian products

Consider vector space A with basis {a1, . . . , ak} Tensor product A ⊗ A is the vector space with basis {ai ⊗ aj}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

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SLIDE 88

Homology of Cartesian products

Consider vector space A with basis {a1, . . . , ak} Tensor product A ⊗ A is the vector space with basis {ai ⊗ aj}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 ⊗ Id2 +2∂ ⊗ ∂ + Id2 ⊗∂2 = 0

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 30 / 37

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SLIDE 89

Homology of Cartesian products

Consider vector space A with basis {a1, . . . , ak} Tensor product A ⊗ A is the vector space with basis {ai ⊗ aj}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 ⊗ Id2 +2∂ ⊗ ∂ + Id2 ⊗∂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

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SLIDE 90

Homology of Cartesian products

Consider vector space A with basis {a1, . . . , ak} Tensor product A ⊗ A is the vector space with basis {ai ⊗ aj}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 ⊗ Id2 +2∂ ⊗ ∂ + Id2 ⊗∂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

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SLIDE 91

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

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SLIDE 92

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

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SLIDE 93

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

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SLIDE 94

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

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SLIDE 95

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

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SLIDE 96

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

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SLIDE 97

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

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SLIDE 98

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

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SLIDE 99

Homotopy invariance

∆H [t1] = [∆t1] = [v ⊗ t1 + t1 ⊗ v] = [v] ⊗ [t1] + [t1] ⊗ [v]

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 33 / 37

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SLIDE 100

Homotopy invariance

∆H [t1] = [∆t1] = [v ⊗ t1 + t1 ⊗ v] = [v] ⊗ [t1] + [t1] ⊗ [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

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SLIDE 101

Homotopy invariance

∆H [t1] = [∆t1] = [v ⊗ t1 + t1 ⊗ v] = [v] ⊗ [t1] + [t1] ⊗ [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]

Ah ha!!! ∆H [t1] = ∆H [t

1] implies UN LN!

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 33 / 37

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SLIDE 102

Homotopy invariance

∆H [t1] = [∆t1] = [v ⊗ t1 + t1 ⊗ v] = [v] ⊗ [t1] + [t1] ⊗ [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]

Ah ha!!! ∆H [t1] = ∆H [t

1] implies UN LN!

And we didn’t know this!! This is a big deal!!!

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 33 / 37

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SLIDE 103

Borromean Rings

If v is a vertex of a connected X and ∆H [c] = [v] ⊗ [c] + [c] ⊗ [v] for all [c] ∈ H (X) , we say that ∆H is primitive

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 34 / 37

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SLIDE 104

Borromean Rings

If v is a vertex of a connected X and ∆H [c] = [v] ⊗ [c] + [c] ⊗ [v] for all [c] ∈ H (X) , we say that ∆H is primitive Theorem (U-M 2010) Kravatz’s diagonal ∆ induces

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 34 / 37

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SLIDE 105

Borromean Rings

If v is a vertex of a connected X and ∆H [c] = [v] ⊗ [c] + [c] ⊗ [v] for all [c] ∈ H (X) , we say that ∆H is primitive Theorem (U-M 2010) Kravatz’s diagonal ∆ induces

a primitive ∆H on H (BR) — the link complement of the Borromean Rings in S3 — indicating that any two of the rings are unlinked

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 34 / 37

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SLIDE 106

Borromean Rings

If v is a vertex of a connected X and ∆H [c] = [v] ⊗ [c] + [c] ⊗ [v] for all [c] ∈ H (X) , we say that ∆H is primitive Theorem (U-M 2010) Kravatz’s diagonal ∆ induces

a primitive ∆H on H (BR) — the link complement of the Borromean Rings in S3 — indicating that any two of the rings are unlinked a non-trivial ∆3 : H (BR) → H (BR)⊗3 indicating that all three rings are linked

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 34 / 37

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SLIDE 107

Tubular neighborhoods of the Borromean Rings

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 35 / 37

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SLIDE 108

Current joint work

with MU undergrad Ben Baer and Dr. Helena Molina-Abril

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 36 / 37

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SLIDE 109

Current joint work

with MU undergrad Ben Baer and Dr. Helena Molina-Abril Conjecture: Kravatz’s diagonal ∆ induces

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 36 / 37

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SLIDE 110

Current joint work

with MU undergrad Ben Baer and Dr. Helena Molina-Abril Conjecture: Kravatz’s diagonal ∆ induces

a primitive ∆H on H (BRn) — the homology of the link complement n Brunian rings in S3 — and trivial maps ∆k : H (BRn) → H (BRn)⊗k for 3 ≤ k ≤ n − 1, indicating that any proper subset of the n Brunian rings is unlinked

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 36 / 37

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SLIDE 111

Current joint work

with MU undergrad Ben Baer and Dr. Helena Molina-Abril Conjecture: Kravatz’s diagonal ∆ induces

a primitive ∆H on H (BRn) — the homology of the link complement n Brunian rings in S3 — and trivial maps ∆k : H (BRn) → H (BRn)⊗k for 3 ≤ k ≤ n − 1, indicating that any proper subset of the n Brunian rings is unlinked a non-trivial ∆n : H (BRn) → H (BRn)⊗n indicating that the n rings are linked

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 36 / 37

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SLIDE 112

Current joint work

with MU undergrad Ben Baer and Dr. Helena Molina-Abril Conjecture: Kravatz’s diagonal ∆ induces

a primitive ∆H on H (BRn) — the homology of the link complement n Brunian rings in S3 — and trivial maps ∆k : H (BRn) → H (BRn)⊗k for 3 ≤ k ≤ n − 1, indicating that any proper subset of the n Brunian rings is unlinked a non-trivial ∆n : H (BRn) → H (BRn)⊗n indicating that the n rings are linked

Remark: If true, we will have found an infinite family of finite dimensional “A∞-coalgebras”, of which few such examples are known

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 36 / 37

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SLIDE 113

The End

Thank you!

(MU/F&M Mathematics Colloquium ) Homology? It’s Mickey Mouse! April 4, 2013 37 / 37