Abstract simplical complexes An abstract simplical complex is a S ′ collection S of finite nonempty subsets. If A ∈ S , then each nonempty subset of A is also in S Example: A ′ a simplex in S ′ ( x, y ) ∈ A ′ and A ′ ∈ S ′ , so ( x, y ) ∈ S ′ The same goes for ( y, z ) etc. x y A ′ z 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 A map is a function that "maps" a point in one space to the other Two abstract complexes S, T are isomorphic if there is a map f mapping the vertices of S to vertices of T
Abstract simplical complexes A map is a function that "maps" a point in one space to the other Two abstract complexes S, T are isomorphic if there is a map f mapping the vertices of S to vertices of T Preserves properties The gray holes are preserved
Abstract simplical complexes A map is a function that "maps" a point in one space to the other 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 { a 0 , ..., a n } ∈ S to { f ( a 0 ) , ..., f ( a n ) } ∈ T
Abstract simplical complexes A map is a function that "maps" a point in one space to the other 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 { a 0 , ..., a n } ∈ S to { f ( a 0 ) , ..., f ( a n ) } ∈ T Assume f is surjective: each point in T has as least one corresponding point in S
Abstract simplical complexes A map is a function that "maps" a point in one space to the other 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 { a 0 , ..., a n } ∈ S to { f ( a 0 ) , ..., f ( a n ) } ∈ T Assume f is surjective: each point in T has as least one corresponding point in S 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
Abstract simplical complexes f d e f Example of a vertex scheme L a b c a
Abstract simplical complexes f d e f Example of a vertex scheme L a b c a e d K f c b a
Abstract simplical complexes f d e f Example of a vertex scheme L a b c a e d K f c b this is a (hollow) cylinder a
Abstract simplical complexes d e f a What will this be? a b c d
Abstract simplical complexes d e f a What will this be? a b c d
Abstract simplical complexes How about this? A B B A
Abstract simplical complexes How about this? A B B A A B B
Abstract simplical complexes How about this? A B B A A B B A It’s a torus! B
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) Faces ADG and CFI do not exist since A and C are on the boundary, unlike DEF 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
Continuing simplical complexes v 2 v 0 v 1
Continuing simplical complexes v 2 v 0 v 1
Continuing simplical complexes v 2 St ( v 0 ) v 0 v 1 The star St ( v ) is the union of interiors of the simplices containing v as vertex
Continuing simplical complexes v 2 St ( v 0 ) v 0 v 1 The star St ( v ) is the union of interiors of the simplices containing v as vertex St ( v ) might not be closed. The closed star St ( v ) is the star of v with the missing edges added.
Continuing simplical complexes v 2 St ( v 0 ) v v 0 Lk ( v ) v 1 The star St ( v ) is the union of interiors of the simplices containing v as vertex 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 )
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 t i v i f can be extended to a map g : | K | → | L | by: x = � n i =0 t i v i ⇒ g ( x ) = � n i =0 t i f ( v i ) f ( v 0 ) , ..., f ( v n ) 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 t i v i f can be extended to a map g : | K | → | L | by: x = � n i =0 t i v i ⇒ g ( x ) = � n i =0 t i f ( v i ) f ( v 0 ) , ..., f ( v n ) 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 t i v i f can be extended to a map g : | K | → | L | by: x = � n i =0 t i v i ⇒ g ( x ) = � n i =0 t i f ( v i ) f ( v 0 ) , ..., f ( v n ) 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 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
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 t i v i f can be extended to a map g : | K | → | L | by: x = � n i =0 t i v i ⇒ g ( x ) = � n i =0 t i f ( v i ) f ( v 0 ) , ..., f ( v n ) 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 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 ( 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
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 t i v i f can be extended to a map g : | K | → | L | by: x = � n i =0 t i v i ⇒ g ( x ) = � n i =0 t i f ( v i ) f ( v 0 ) , ..., f ( v n ) 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 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 ( 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 g is called a simplical map
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 l m k K K H I n j G h i J L o F h E M L g N (empty) D a b c d e f What is St ( j ) and h ( St ( j )) ? B O C A K is a circle, L is a annulus
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 l m k K K H I n j G h i J L o F h E M L g N (empty) D a b c d e f What is St ( j ) and h ( St ( j )) ? Where is w ? B O C A K is a circle, L is a annulus
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 l w St ( w ) m k K K H I n j G h i J L o F h E M L g N (empty) D a b c d e f What is St ( j ) and h ( St ( j )) ? Where is w ? B O C A K is a circle, L is a annulus
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 l w St ( w ) m k K K H I n j G h i J L o F h E M L g (empty) D a b c d e f N What is St ( j ) and h ( St ( j )) ? Where is w ? B O It follows that h ( St ( j )) ⊆ St ( w ) C A K is a circle, L is a annulus
Simplical approximations Let h : | K | → | L | be a continuous map Choose f : K (0) → L (0) L L K K I = J I = J l K G G m k n j i f F = H F = H o h L L g E E M M (empty) a b c d e f N = O N = O C = D C = D A = B A = B
Simplical approximations Let h : | K | → | L | be a continuous map Choose f : K (0) → L (0) L L K K I = J I = J l K G G m k n j i f F = H F = H o h L L g E E M M (empty) a b c d e f N = O N = O C = D C = D A = B A = B f is a simplical approximation to h if for each vertex v in K h ( St ( v )) ⊆ St ( f ( v )) holds
Simplical approximations Let h : | K | → | L | be a continuous map Choose f : K (0) → L (0) L L K K I = J I = J l K G G m k n j i f F = H F = H o h L L g E E M M (empty) a b c d e f N = O N = O C = D C = D St ( j ) A = B A = B f is a simplical approximation to h if for each vertex v in K h ( St ( v )) ⊆ St ( f ( v )) holds
Simplical approximations Let h : | K | → | L | be a continuous map Choose f : K (0) → L (0) L L K K I = J I = J l K G G m k n j i f F = H F = H o h L L g E E M M (empty) a b c d e f N = O N = O C = D C = D St ( j ) St ( f ( j )) = St ( J ) A = B A = B f is a simplical approximation to h if for each vertex v in K h ( St ( v )) ⊆ St ( f ( v )) holds
Simplical approximations Let h : | K | → | L | be a continuous map Choose f : K (0) → L (0) L L K K I = J I = J l K G G m k K ′ I ′ J ′ n j i f F = H F = H o h L L g E E M M (empty) a b c d e f N = O N = O C = D C = D St ( j ) St ( f ( j )) = St ( J ) h ( j ) is the set of green line segments A = B A = B f is a simplical approximation to h if for each vertex v in K h ( St ( v )) ⊆ St ( f ( v )) holds
Simplical approximations Let h : | K | → | L | be a continuous map Choose f : K (0) → L (0) So f is an simplical approximation of h L L K K I = J I = J l K G G m k K ′ I ′ J ′ n j i f F = H F = H o h L L g E E M M (empty) a b c d e f N = O N = O C = D C = D St ( j ) St ( f ( j )) = St ( J ) h ( j ) is the set of green line segments A = B A = B f is a simplical approximation to h if for each vertex v in K h ( St ( v )) ⊆ St ( f ( v )) holds
Simplical approximation theorem Some definitions: σ = � p 1 A barycenter of σ is defined as a point: ˆ p +1 v i i =0
Simplical approximation theorem Some definitions: σ = � p 1 A barycenter of σ is defined as a point: ˆ p +1 v i i =0 It is the point in Int ( σ ) where all the barycentric coordinates are equal ˆ σ ˆ σ
Simplical approximation theorem Some definitions: σ = � p 1 A barycenter of σ is defined as a point: ˆ p +1 v i i =0 It is the point in Int ( σ ) where all the barycentric coordinates are equal K ˆ σ Let L 0 = K (0)
Simplical approximation theorem Some definitions: σ = � p 1 A barycenter of σ is defined as a point: ˆ p +1 v i i =0 It is the point in Int ( σ ) where all the barycentric coordinates are equal First iteration K ˆ s ˆ σ L 1 is the subdivision of L 0 . It is obtained by using the barycenters of L 0
Simplical approximation theorem Some definitions: σ = � p 1 A barycenter of σ is defined as a point: ˆ p +1 v i i =0 It is the point in Int ( σ ) where all the barycentric coordinates are equal First iteration K Second iteration ˆ s ˆ σ L 2 is the subdivision of L 1 . It is obtained by using the barycenters of L 1 which is obtaind by using the barycenters of L 0
Simplical approximation theorem Some definitions: σ = � p 1 A barycenter of σ is defined as a point: ˆ p +1 v i i =0 It is the point in Int ( σ ) where all the barycentric coordinates are equal First iteration sd ( σ ) K sd 2 ( σ ) Second iteration ˆ s ˆ σ sd n ( σ ) Etc. This is called a barycentric subdivision sd ( σ ) L n is the subdivision of L n − 1 . It is obtained by using the barycenters of L n − 1 which is obtaind by using the barycenters of L n − 2 which... etc.
Simplical approximation theorem Some definitions: σ = � p 1 A barycenter of σ is defined as a point: ˆ p +1 v i i =0 It is the point in Int ( σ ) where all the barycentric coordinates are equal First iteration sd ( σ ) K sd 2 ( σ ) Second iteration ˆ s ˆ σ sd n ( σ ) Etc. This is called a barycentric subdivision sd ( σ ) L n is the subdivision of L n − 1 . It is obtained by using the barycenters of L n − 1 which is obtaind by using the barycenters of L n − 2 which... etc. Splits each n -simplex into smaller ( n − 1) -simplices
Simplical approximation theorem l K K I H G J L F M E (empty) L N g a D B C O A h
Simplical approximation theorem l K K I H G J L F M E (empty) L N g a D B C O A h h violations of the star condition: for each edge in K , there is no w ∈ L (0)
Simplical approximation theorem l K K I H G J L F M E (empty) L N g a D B C O A h h violations of the star condition: for each edge in K , there is no w ∈ L (0) So we make subdivisions of K until it the condition is satisfied
Simplical approximation theorem l K K I H G J L F M E (empty) L N g a a ′ D B C O A h h violations of the star condition: for each edge in K , there is no w ∈ L (0) ( a, a ′ ) is still a violation
Simplical approximation theorem l K K I H G J L F M E (empty) L N g a a ′ D B C O A h h violations of the star condition: for each edge in K , there is no w ∈ L (0) ( a, a ′ ) is still a violation
Simplical approximation theorem l K K I H G J L F M E (empty) L N g a a ′′ a ′ D B C O A h h violations of the star condition: for each edge in K , there is no w ∈ L (0) ( a, a ′ ) is still a violation ( a, a ′′ ) is not
Simplical approximation theorem l K K I H G J L F M E (empty) L N g a a ′′ a ′ D B C O A h h violations of the star condition: for each edge in K , there is no w ∈ L (0) ( a, a ′ ) is still a violation ( a, a ′′ ) is not h : K → L has a simplical approximation f : sd 2 ( K ) → L
Simplical approximation theorem This can be generalized to the 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 : sd N ( K ) → L
Simplical approximation theorem This can be generalized to the 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 : sd N ( K ) → L Proof: cover | K | with open sets h − 1 ( St ( v )) , v ∈ L
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