Simplical complexes Bas van Loon b.v.loon@student.tue.nl - - PowerPoint PPT Presentation

Bas van Loon b.v.loon@student.tue.nl

Simplical complexes

Introduction

Topics: simplices, complexes, maps, simplical approximations, nerve thereom, ...

Introduction

Topics: simplices, complexes, maps, simplical approximations, nerve thereom, ... More details in the literature

Simplices

σ is an n-simplex "spanned" between the points in P where n is the number of dimensions, the number of vertices of σ is n + 1

Simplices

σ is an n-simplex "spanned" between the points in P where n is the number of dimensions, the number of vertices of σ is n + 1

a0 a0 a1 a0 a1 a2 a0 a3 a1 a2

d = 0 d = 1 d = 2 d = 3

Simplices

σ is an n-simplex "spanned" between the points in P where n is the number of dimensions, the number of vertices of σ is n + 1

a0 a0 a1 a0 a1 a2 a0 a3 a1 a2

spanned means: "connect all the vertices" d = 0 d = 1 d = 2 d = 3

Simplices

σ is an n-simplex "spanned" between the points in P where n is the number of dimensions, the number of vertices of σ is n + 1

a0 a0 a1 a0 a1 a2 a0 a3 a1 a2

spanned means: "connect all the vertices" Also possible for n > 3 d = 0 d = 1 d = 2 d = 3

Simplices v0 v1 v0 v1 v2

Simplices

O

v0 v1 v0 v1 v2

Simplices

O

v0 v1 v0 v1 v2 v′ = 1

2v0 + 1 2v1

v′

Simplices

O O

v0 v1 v0 v1 v2 v′ = 1

2v0 + 1 2v1

v′

Simplices

O O

v0 v1 v0 v1 v2 v′ = 1

2v0 + 1 2v1

v′ v′ = 0v0 + 1

3v1 + 2 3v2

Simplices

O O

v0 v1 v0 v1 v2 v′ = 1

2v0 + 1 2v1

v′ v′ v′ = 1

3v0 + 1 3v1 + 1 3v2

Simplices

Barycentric coordinates are defined as a set of points x in Rn space such that each point in x is defined as n

i=0 tiai where n i=0 ti = 1

and each ti ≥ 0.

Simplices

Barycentric coordinates are defined as a set of points x in Rn space such that each point in x is defined as n

i=0 tiai where n i=0 ti = 1

and each ti ≥ 0. σ is defined as the set of all points x ∈ Rn such that: x = n

i=0 tiai,

where n

i=0 ti = 1 for all ti ≥ 0

σ is convex and compact e.g. σ has edges like [a, b], not (a, b) (or: closed)

Simplices

Barycentric coordinates are defined as a set of points x in Rn space such that each point in x is defined as n

i=0 tiai where n i=0 ti = 1

and each ti ≥ 0. σ is defined as the set of all points x ∈ Rn such that: x = n

i=0 tiai,

where n

i=0 ti = 1 for all ti ≥ 0

Any simplex spanned by a subset of {a0, ..., an} is called a face. The face spanned by {a1, ..., an} is called the face opposite a0.

a0 a1 a2 a3 a0 a1 a2 Simplices

Barycentric coordinates are defined as a set of points x in Rn space such that each point in x is defined as n

i=0 tiai where n i=0 ti = 1

and each ti ≥ 0. σ is defined as the set of all points x ∈ Rn such that: x = n

i=0 tiai,

where n

i=0 ti = 1 for all ti ≥ 0

Proper faces are faces of σ not equal to σ. The union of the proper faces is called the boundary of σ and is denoted as Bd(σ)

a0 a1 a2 a3 a0 a1 a2 Simplices

Proper faces are faces of σ not equal to σ. The union of the proper faces is called the boundary of σ and is denoted as Bd(σ) The interior or open simplex of σ is defined as Int(σ) = σ − Bd(σ)

a0 a1 a2 a3 a0 a1 a2 Simplices

Proper faces are faces of σ not equal to σ. The union of the proper faces is called the boundary of σ and is denoted as Bd(σ) The interior or open simplex of σ is defined as Int(σ) = σ − Bd(σ) x ∈ Bd(σ) if at least one ti(x) = 0 x ∈ Int(σ) if for all ti(x) > 0

a0 a1 a2 a3 a0 a1 a2 Simplices

Simplical complex

A simplical complex K in Rn is a collection of simplices in Rn

Examples:

Simplical complex

A simplical complex K in Rn is a collection of simplices in Rn

Examples:

Simplical complex

A simplical complex K in Rn is a collection of simplices in Rn 1) Every face of a simplex of K is in K 2) Every pair of distinct simplices of K has a disjoint interior 3) The intersection of any two distinct simplices of K is a face of each of them

Examples: a0 a1 a2 a3 a4

If L is a subcollection of K, then L is a subcomplex on its own: {a1, a2, a4}

Simplical complex

A simplical complex K in Rn is a collection of simplices in Rn 1) Every face of a simplex of K is in K 2) Every pair of distinct simplices of K has a disjoint interior 3) The intersection of any two distinct simplices of K is a face of each of them

Examples: a0 a1 a2 a3 a4

If L is a subcollection of K, then L is a subcomplex on its own: {a1, a2, a4} A subcollection or p-skeleton K(p) is the collection of p-simplices in K

Simplical complex

A simplical complex K in Rn is a collection of simplices in Rn 1) Every face of a simplex of K is in K 2) Every pair of distinct simplices of K has a disjoint interior 3) The intersection of any two distinct simplices of K is a face of each of them

Examples: a0 a1 a2 a3 a4

If L is a subcollection of K, then L is a subcomplex on its own: {a1, a2, a4} A subcollection or p-skeleton K(p) is the collection of p-simplices in K K(0) are the vertices of K

Simplical complex

A simplical complex K in Rn is a collection of simplices in Rn 1) Every face of a simplex of K is in K 2) Every pair of distinct simplices of K has a disjoint interior 3) The intersection of any two distinct simplices of K is a face of each of them

Examples: a0 a1 a2 a3 a4

If L is a subcollection of K, then L is a subcomplex on its own: {a1, a2, a4} A subcollection or p-skeleton K(p) is the collection of p-simplices in K K(0) are the vertices of K K(1) are the edges of K

Simplical complex

A simplical complex K in Rn is a collection of simplices in Rn 1) Every face of a simplex of K is in K 2) Every pair of distinct simplices of K has a disjoint interior 3) The intersection of any two distinct simplices of K is a face of each of them

Examples: a0 a1 a2 a3 a4

If L is a subcollection of K, then L is a subcomplex on its own: {a1, a2, a4} A subcollection or p-skeleton K(p) is the collection of p-simplices in K K(0) are the vertices of K K(1) are the edges of K K(2) are the faces of K, etc.

Simplical complex

|K| is a collection of the union of simplices in K

Simplical complex

|K| is a collection of the union of simplices in K |K| is called the polytope or polyhedron or underlying space of K

Simplical complex

|K| is a collection of the union of simplices in K |K| is called the polytope or polyhedron or underlying space of K The topology of |K| can be larger than the topology of K

Simplical complex

|K| is a collection of the union of simplices in K |K| is called the polytope or polyhedron or underlying space of K The topology of |K| can be larger than the topology of K Example: K is the set of 1-simplices in dimension R [m, m + 1] and [

1 n+1, 1 n]

for integers n, m > 0

Simplical complex

|K| is a collection of the union of simplices in K |K| is called the polytope or polyhedron or underlying space of K The topology of |K| can be larger than the topology of K Example: K is the set of 1-simplices in dimension R [m, m + 1] and [

1 n+1, 1 n]

for integers n, m > 0 The underlying space |K| = (0, 1] [1, ∞) = (0, ∞)

Simplical complex

|K| is a collection of the union of simplices in K |K| is called the polytope or polyhedron or underlying space of K The topology of |K| can be larger than the topology of K Example: K is the set of 1-simplices in dimension R [m, m + 1] and [

1 n+1, 1 n]

for integers n, m > 0 The underlying space |K| = (0, 1] [1, ∞) = (0, ∞) The set of points defined as 1

n is closed in |K| and not closed in any

σ ∈ K

Abstract simplical complexes

Abstract simplical complexes

An abstract simplical complex is a collection S of finite nonempty subsets. S′

Abstract simplical complexes

An abstract simplical complex is a collection S of finite nonempty subsets. If A ∈ S, then each nonempty subset

• f A is also in S

A′ S′

Abstract simplical complexes

An abstract simplical complex is a collection S of finite nonempty subsets. If A ∈ S, then each nonempty subset

• f A is also in S

A′ S′ Example: A′ a simplex in S′

Abstract simplical complexes

An abstract simplical complex is a collection S of finite nonempty subsets. If A ∈ S, then each nonempty subset

• f A is also in S

A′ (x, y) ∈ A′ and A′ ∈ S′, so (x, y) ∈ S′ x y S′ z Example: A′ a simplex in S′

Abstract simplical complexes

An abstract simplical complex is a collection S of finite nonempty subsets. If A ∈ S, then each nonempty subset

• f A is also in S

A′ (x, y) ∈ A′ and A′ ∈ S′, so (x, y) ∈ S′ x y S′ z Example: A′ a simplex in S′ The same goes for (y, z) etc.

Abstract simplical complexes

An abstract simplical complex is a collection S of finite nonempty subsets. If A ∈ S, then each nonempty subset

• f A is also in S

A′ (x, y) ∈ A′ and A′ ∈ S′, so (x, y) ∈ S′ x y S′ z Example: A′ a simplex in S′ The same goes for (y, z) etc. The subcomplexes of simplices A ∈ C (vertices, edges, faces, tetrahedra, etc.) are all elements of S.

Abstract simplical complexes

A map is a function that "maps" a point in one space to the other

Abstract simplical complexes

Two abstract complexes S, T are isomorphic if there is a map f mapping the vertices of S to vertices of T A map is a function that "maps" a point in one space to the other

Abstract simplical complexes

Two abstract complexes S, T are isomorphic if there is a map f mapping the vertices of S to vertices of T Preserves properties A map is a function that "maps" a point in one space to the other

The gray holes are preserved

Abstract simplical complexes

Two abstract complexes S, T are isomorphic if there is a map f mapping the vertices of S to vertices of T Preserves properties Maps {a0, ..., an} ∈ S to {f(a0), ..., f(an)} ∈ T A map is a function that "maps" a point in one space to the other

Abstract simplical complexes

Two abstract complexes S, T are isomorphic if there is a map f mapping the vertices of S to vertices of T Preserves properties Maps {a0, ..., an} ∈ S to {f(a0), ..., f(an)} ∈ T Assume f is surjective: each point in T has as least one corresponding point in S A map is a function that "maps" a point in one space to the other

Abstract simplical complexes

Two abstract complexes S, T are isomorphic if there is a map f mapping the vertices of S to vertices of T Preserves properties Maps {a0, ..., an} ∈ S to {f(a0), ..., f(an)} ∈ T K is a simplical complex with vertex set V . K is the collection of all subsets of V and is called the vertex scheme of K Assume f is surjective: each point in T has as least one corresponding point in S A map is a function that "maps" a point in one space to the other

Abstract simplical complexes

L f a b c a d e f Example of a vertex scheme

Abstract simplical complexes

L f a b c a d e f K a b d

e

f c Example of a vertex scheme

Abstract simplical complexes

L f a b c a d e f K a b d

e

f c this is a (hollow) cylinder Example of a vertex scheme

Abstract simplical complexes

a a d d c b e f What will this be?

Abstract simplical complexes

a a d d c b e f What will this be?

Abstract simplical complexes

B B A A How about this?

Abstract simplical complexes

B B A A A B B How about this?

Abstract simplical complexes

B B A A A B B A B How about this? It’s a torus!

Abstract simplical complexes

Another vertex map (torus)

Abstract simplical complexes

Another vertex map (torus)

Abstract simplical complexes

Another vertex map (torus)

Abstract simplical complexes

Another vertex map (torus) C has edges to A and B since ABC is a face. C has also edges to I and F hence a hole is created (with ADG)

Abstract simplical complexes

Another vertex map (torus) C has edges to A and B since ABC is a face. C has also edges to I and F hence a hole is created (with ADG)

Abstract simplical complexes

Another vertex map (torus) C has edges to A and B since ABC is a face. C has also edges to I and F hence a hole is created (with ADG) Edges (E, H), (H, B), (B, E) form the edges of the outside of the torus

Abstract simplical complexes

Another vertex map (torus) C has edges to A and B since ABC is a face. C has also edges to I and F hence a hole is created (with ADG) Edges (E, H), (H, B), (B, E) form the edges of the outside of the torus Faces ADG and CFI do not exist since A and C are on the boundary, unlike DEF

Continuing simplical complexes

v0 v1 v2

Continuing simplical complexes

v0 v1 v2

Continuing simplical complexes

The star St(v) is the union of interiors of the simplices containing v as vertex v0 v1 v2 St(v0)

Continuing simplical complexes

The star St(v) is the union of interiors of the simplices containing v as vertex v0 v1 v2 St(v0) St(v) might not be closed. The closed star St(v) is the star of v with the missing edges added.

Continuing simplical complexes

The star St(v) is the union of interiors of the simplices containing v as vertex v0 v1 v2 St(v0) St(v) might not be closed. The closed star St(v) is the star of v with the missing edges added. The link of v is defined as Lk(v) = St(v) − St(v) Lk(v) v

Continuing simplical complexes

Let f be a map f : K(0) → L(0)

Continuing simplical complexes

Let f be a map f : K(0) → L(0) Remember: a point in a simplex is defined as x = n

i=0 tivi

f can be extended to a map g : |K| → |L| by: x = n

i=0 tivi ⇒ g(x) = n i=0 tif(vi)

f(v0), ..., f(vn) might not be distinct, but still span a simplex τ of L

Continuing simplical complexes

Let f be a map f : K(0) → L(0) Remember: a point in a simplex is defined as x = n

i=0 tivi

f can be extended to a map g : |K| → |L| by: x = n

i=0 tivi ⇒ g(x) = n i=0 tif(vi)

f(v0), ..., f(vn) might not be distinct, but still span a simplex τ of L f maps a simplex σ in K to a simplex τ in L

Continuing simplical complexes

Let f be a map f : K(0) → L(0) Remember: a point in a simplex is defined as x = n

i=0 tivi

f can be extended to a map g : |K| → |L| by: x = n

i=0 tivi ⇒ g(x) = n i=0 tif(vi)

f(v0), ..., f(vn) might not be distinct, but still span a simplex τ of L The coefficients of g(x) are non-negative and the sum is 1. So, g(x) is a point in τ and g(x) is continuous from σ to τ f maps a simplex σ in K to a simplex τ in L

Continuing simplical complexes

Let f be a map f : K(0) → L(0) Remember: a point in a simplex is defined as x = n

i=0 tivi

f can be extended to a map g : |K| → |L| by: x = n

i=0 tivi ⇒ g(x) = n i=0 tif(vi)

f(v0), ..., f(vn) might not be distinct, but still span a simplex τ of L g(x) is a continuous map from σ to τ and hence a map into L. The map |K| → |L| is continuous since g(x) is also continuous The coefficients of g(x) are non-negative and the sum is 1. So, g(x) is a point in τ and g(x) is continuous from σ to τ f maps a simplex σ in K to a simplex τ in L

Continuing simplical complexes

Let f be a map f : K(0) → L(0) Remember: a point in a simplex is defined as x = n

i=0 tivi

f can be extended to a map g : |K| → |L| by: x = n

i=0 tivi ⇒ g(x) = n i=0 tif(vi)

f(v0), ..., f(vn) might not be distinct, but still span a simplex τ of L g(x) is a continuous map from σ to τ and hence a map into L. The map |K| → |L| is continuous since g(x) is also continuous The coefficients of g(x) are non-negative and the sum is 1. So, g(x) is a point in τ and g(x) is continuous from σ to τ g is called a simplical map f maps a simplex σ in K to a simplex τ in L

Simplical approximations

Let h : |K| → |L| be a continuous map

Simplical approximations

Let h : |K| → |L| be a continuous map The star condition says that for all vertices v in K, a vertex w in L exists such sthat h(St(v)) ⊆ St(w). Or: each vertex star in K is contained in L

Simplical approximations

Let h : |K| → |L| be a continuous map The star condition says that for all vertices v in K, a vertex w in L exists such sthat h(St(v)) ⊆ St(w). Or: each vertex star in K is contained in L h a b c d e f g h i j k l m n

• K

L K is a circle, L is a annulus What is St(j) and h(St(j))? O A B C D E F G H I J K L M N (empty)

Simplical approximations

Let h : |K| → |L| be a continuous map The star condition says that for all vertices v in K, a vertex w in L exists such sthat h(St(v)) ⊆ St(w). Or: each vertex star in K is contained in L h a b c d e f g h i j k l m n

• K

L K is a circle, L is a annulus What is St(j) and h(St(j))? Where is w? O A B C D E F G H I J K L M N (empty)

Simplical approximations

Let h : |K| → |L| be a continuous map The star condition says that for all vertices v in K, a vertex w in L exists such sthat h(St(v)) ⊆ St(w). Or: each vertex star in K is contained in L h a b c d e f g h i j k l m n

• K

L K is a circle, L is a annulus What is St(j) and h(St(j))? Where is w? w St(w) O A B C D E F G H I J K L M N (empty)

Simplical approximations

Let h : |K| → |L| be a continuous map The star condition says that for all vertices v in K, a vertex w in L exists such sthat h(St(v)) ⊆ St(w). Or: each vertex star in K is contained in L h a b c d e f g h i j k l m n

• K

L K is a circle, L is a annulus What is St(j) and h(St(j))? Where is w? w St(w) O A B C D E F G H I J K L M N It follows that h(St(j)) ⊆ St(w) (empty)

Simplical approximations

Let h : |K| → |L| be a continuous map Choose f : K(0) → L(0) a b c d e g h i j k l m n

• K

f A = B N = O C = D E F = H G I = J L K M

L

f A = B N = O C = D E F = H G I = J L K M

L

(empty)

Simplical approximations

Let h : |K| → |L| be a continuous map f is a simplical approximation to h if for each vertex v in K h(St(v)) ⊆ St(f(v))holds Choose f : K(0) → L(0) a b c d e g h i j k l m n

• K

f A = B N = O C = D E F = H G I = J L K M

L

f A = B N = O C = D E F = H G I = J L K M

L

(empty)

Simplical approximations

Let h : |K| → |L| be a continuous map f is a simplical approximation to h if for each vertex v in K h(St(v)) ⊆ St(f(v))holds Choose f : K(0) → L(0) a b c d e g h i j k l m n

• K

f A = B N = O C = D E F = H G I = J L K M

L

St(j) f A = B N = O C = D E F = H G I = J L K M

L

(empty)

Simplical approximations

Let h : |K| → |L| be a continuous map f is a simplical approximation to h if for each vertex v in K h(St(v)) ⊆ St(f(v))holds Choose f : K(0) → L(0) a b c d e g h i j k l m n

• K

f A = B N = O C = D E F = H G I = J L K M

L

St(j) f St(f(j)) = St(J) A = B N = O C = D E F = H G I = J L K M

L

(empty)

Simplical approximations

Let h : |K| → |L| be a continuous map f is a simplical approximation to h if for each vertex v in K h(St(v)) ⊆ St(f(v))holds Choose f : K(0) → L(0) a b c d e g h i j k l m n

• K

f A = B N = O C = D E F = H G I = J L K M

L

St(j) f St(f(j)) = St(J) A = B N = O C = D E F = H G I = J L K M

L

(empty) h(j) is the set of green line segments K′ J′ I′

Simplical approximations

Let h : |K| → |L| be a continuous map f is a simplical approximation to h if for each vertex v in K h(St(v)) ⊆ St(f(v))holds Choose f : K(0) → L(0) a b c d e g h i j k l m n

• K

f A = B N = O C = D E F = H G I = J L K M

L

St(j) f St(f(j)) = St(J) A = B N = O C = D E F = H G I = J L K M

L

(empty) h(j) is the set of green line segments So f is an simplical approximation of h K′ J′ I′

Simplical approximation theorem

A barycenter of σ is defined as a point: ˆ σ = p

i=0 1 p+1vi

Some definitions:

Simplical approximation theorem

A barycenter of σ is defined as a point: ˆ σ = p

i=0 1 p+1vi

Some definitions: It is the point in Int(σ) where all the barycentric coordinates are equal ˆ σ ˆ σ

Simplical approximation theorem

A barycenter of σ is defined as a point: ˆ σ = p

i=0 1 p+1vi

Some definitions: It is the point in Int(σ) where all the barycentric coordinates are equal ˆ σ Let L0 = K(0)

K

Simplical approximation theorem

A barycenter of σ is defined as a point: ˆ σ = p

i=0 1 p+1vi

Some definitions: It is the point in Int(σ) where all the barycentric coordinates are equal ˆ σ First iteration ˆ s

K

L1 is the subdivision of L0. It is obtained by using the barycenters of L0

Simplical approximation theorem

A barycenter of σ is defined as a point: ˆ σ = p

i=0 1 p+1vi

Some definitions: It is the point in Int(σ) where all the barycentric coordinates are equal ˆ σ First iteration Second iteration ˆ s

K

L2 is the subdivision of L1. It is obtained by using the barycenters of L1 which is obtaind by using the barycenters of L0

Simplical approximation theorem

This is called a barycentric subdivision sd(σ) A barycenter of σ is defined as a point: ˆ σ = p

i=0 1 p+1vi

Some definitions: It is the point in Int(σ) where all the barycentric coordinates are equal ˆ σ First iteration Second iteration ˆ s Etc. sd(σ) sd2(σ) sdn(σ)

K

Ln is the subdivision of Ln−1. It is obtained by using the barycenters of Ln−1 which is obtaind by using the barycenters of Ln−2 which... etc.

Simplical approximation theorem

This is called a barycentric subdivision sd(σ) A barycenter of σ is defined as a point: ˆ σ = p

i=0 1 p+1vi

Some definitions: It is the point in Int(σ) where all the barycentric coordinates are equal ˆ σ First iteration Second iteration ˆ s Etc. sd(σ) sd2(σ) sdn(σ)

K

Ln is the subdivision of Ln−1. It is obtained by using the barycenters of Ln−1 which is obtaind by using the barycenters of Ln−2 which... etc. Splits each n-simplex into smaller (n − 1)-simplices

Simplical approximation theorem

L

O A B C D E F G H I J K L M N (empty)

K

a g l h

Simplical approximation theorem

L

O A B C D E F G H I J K L M N (empty) h violations of the star condition: for each edge in K, there is no w ∈ L(0)

K

a g l h

Simplical approximation theorem

L

O A B C D E F G H I J K L M N (empty) h violations of the star condition: for each edge in K, there is no w ∈ L(0)

K

a g l h So we make subdivisions of K until it the condition is satisfied

Simplical approximation theorem

L

O A B C D E F G H I J K L M N (empty) h violations of the star condition: for each edge in K, there is no w ∈ L(0)

K

a g l h a′ (a, a′) is still a violation

Simplical approximation theorem

L

O A B C D E F G H I J K L M N (empty) h violations of the star condition: for each edge in K, there is no w ∈ L(0)

K

a g l h a′ (a, a′) is still a violation

Simplical approximation theorem

L

O A B C D E F G H I J K L M N (empty) h violations of the star condition: for each edge in K, there is no w ∈ L(0)

K

a g l h a′ (a, a′) is still a violation a′′ (a, a′′) is not

Simplical approximation theorem

L

O A B C D E F G H I J K L M N (empty) h violations of the star condition: for each edge in K, there is no w ∈ L(0)

K

a g l h a′ (a, a′) is still a violation a′′ (a, a′′) is not h : K → L has a simplical approximation f : sd2(K) → L

Simplical approximation theorem

Given is: complexes K, L (K is finite) and a continuous map h : |K| → |L| There is a number N such that h has a simplical approximation f : sdN(K) → L This can be generalized to the simplical approximation theorem:

Simplical approximation theorem

Given is: complexes K, L (K is finite) and a continuous map h : |K| → |L| There is a number N such that h has a simplical approximation f : sdN(K) → L This can be generalized to the simplical approximation theorem: Proof: cover |K| with open sets h−1(St(v)), v ∈ L

Simplical approximation theorem

Given is: complexes K, L (K is finite) and a continuous map h : |K| → |L| There is a number N such that h has a simplical approximation f : sdN(K) → L This can be generalized to the simplical approximation theorem: Proof: cover |K| with open sets h−1(St(v)), v ∈ L Example of worst case: λ g−1(St(v)) st(k) A vertex k in K

Simplical approximation theorem

Given is: complexes K, L (K is finite) and a continuous map h : |K| → |L| There is a number N such that h has a simplical approximation f : sdN(K) → L This can be generalized to the simplical approximation theorem: Proof: cover |K| with open sets h−1(St(v)), v ∈ L K is finite (and compact), so there is a λ ≥ 0 such that the diameter

• f h−1(St(v)) < λ

2

Subdivisions become smaller after each iteration

Simplical approximation theorem

Given is: complexes K, L (K is finite) and a continuous map h : |K| → |L| There is a number N such that h has a simplical approximation f : sdN(K) → L This can be generalized to the simplical approximation theorem: Proof: cover |K| with open sets h−1(St(v)), v ∈ L K is finite (and compact), so there is a λ ≥ 0 such that the diameter

• f h−1(St(v)) < λ

2

Subdivisions become smaller after each iteration This means there is an n such that the diameter of each vertex in sdn(K) < λ

2

Simplical approximation theorem

Given is: complexes K, L (K is finite) and a continuous map h : |K| → |L| There is a number N such that h has a simplical approximation f : sdN(K) → L This can be generalized to the simplical approximation theorem: Proof: cover |K| with open sets h−1(St(v)), v ∈ L K is finite (and compact), so there is a λ ≥ 0 such that the diameter

• f h−1(St(v)) < λ

2

Subdivisions become smaller after each iteration This means there is an n such that the diameter of each vertex in sdn(K) < λ

2

This implies that each star in K lies in a set h−1(St(v))

Nerve theorem

A homotopy is a transformation between 2 spaces X, Y

Nerve theorem

A homotopy is a transformation between 2 spaces X, Y X and Y are homotopy equivalent if maps f : X → Y and g : Y → X exist where f ◦ g is homotopic to idX and g ◦ f is homotopic to idY

Nerve theorem

A homotopy is a transformation between 2 spaces X, Y X and Y are homotopy equivalent if maps f : X → Y and g : Y → X exist where f ◦ g is homotopic to idX and g ◦ f is homotopic to idY idX is the identity map of X, f(x) = x

Nerve theorem

A homotopy is a transformation between 2 spaces X, Y X and Y are homotopy equivalent if maps f : X → Y and g : Y → X exist where f ◦ g is homotopic to idX and g ◦ f is homotopic to idY idX is the identity map of X, f(x) = x X and Y have the same homotopy type if maps f ◦ g : X → X and g ◦ f : Y → Y are homotopic

Nerve theorem

A homotopy does not exist in this case torus trefoil knot

Nerve theorem

Given is a set of shapes F

Nerve theorem

A nerve Nrv(F) is a simplical complex such that: Nrv(F) = {X ⊆ F| X = ∅} Given is a set of shapes F

Nerve theorem

A nerve Nrv(F) is a simplical complex such that: Nrv(F) = {X ⊆ F| X = ∅} Given is a set of shapes F

A B C D

E F G

Nerve theorem

A nerve Nrv(F) is a simplical complex such that: Nrv(F) = {X ⊆ F| X = ∅} Given is a set of shapes F

A B C D

E F G

Nerve theorem

The nerve theorem (by Edelsbrunner, Harer) states that the nerves in a set S and the unions in S have the same homotopy type

Nerve theorem

A B C D

The nerve theorem (by Edelsbrunner, Harer) states that the nerves in a set S and the unions in S have the same homotopy type

E F G

S′

a f b g e d c

Nerve theorem

A B C D

The nerve theorem (by Edelsbrunner, Harer) states that the nerves in a set S and the unions in S have the same homotopy type

E F G

S′

a f b g e d c Let f be a homotopy from X ∈ F to its corresponding nerves and let g be a homotopy back to X

Nerve theorem

A B C D

The nerve theorem (by Edelsbrunner, Harer) states that the nerves in a set S and the unions in S have the same homotopy type

E F G

S′

a f b g e d c Let f be a homotopy from X ∈ F to its corresponding nerves and let g be a homotopy back to X f, g f, g

Sources

• J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley

Publishing Company, 1984 http://mathworld.wolfram.com/Homeomorphism.html https://en.wikipedia.org/wiki/Homeomorphism http: //mathworld.wolfram.com/GeometricRealization.html http://mathworld.wolfram.com/Homotopy.html http://mathworld.wolfram.com/HomotopyType.html

• H. Edelsbrunner and J. Harer, Computational topology. An

introduction, Departments of Computer Science and Mathematics of Duke University https://en.wikipedia.org/wiki/Homotopy

Questions