GAP (HAP commands) - Chapter 28: Simplicial Complexes

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28 Simplicial Complexes
28.1
28.1-1 Homology
28.1-2 RipsHomology
28.1-3 Bettinumbers
28.1-4 ChainComplex
28.1-5 CechComplexOfPureCubicalComplex
28.1-6 PureComplexToSimplicialComplex
28.1-7 RipsChainComplex
28.1-8 VectorsToSymmetricMatrix
28.1-9 EulerCharacteristic
28.1-10 MaximalSimplicesToSimplicialComplex
28.1-11 SkeletonOfSimplicialComplex
28.1-12 GraphOfSimplicialComplex
28.1-13 ContractibleSubcomplexOfSimplicialComplex
28.1-14 PathComponentsOfSimplicialComplex
28.1-15 QuillenComplex
28.1-16 SymmetricMatrixToIncidenceMatrix
28.1-17 IncidenceMatrixToGraph
28.1-18 CayleyGraphOfGroup
28.1-19 PathComponentsOfGraph
28.1-20 ContractGraph
28.1-21 GraphDisplay
28.1-22 SimplicialMap
28.1-23 ChainMapOfSimplicialMap
28.1-24 SimplicialNerveOfGraph

### 28 Simplicial Complexes

#### 28.1

##### 28.1-1 Homology

`‣ Homology`( T, n ) | ( function )

`‣ Homology`( T ) | ( function )

Inputs a pure cubical complex, or cubical complex, or simplicial complex T and a non-negative integer n. It returns the n-th integral homology of T as a list of torsion integers. If no value of n is input then the list of all homologies of T in dimensions 0 to Dimension(T) is returned .

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41

##### 28.1-2 RipsHomology

`‣ RipsHomology`( G, n ) | ( function )

`‣ RipsHomology`( G, n, p ) | ( function )

Inputs a graph G, a non-negative integer n (and optionally a prime number p). It returns the integral homology (or mod p homology) in degree n of the Rips complex of G.

Examples:

##### 28.1-3 Bettinumbers

`‣ Bettinumbers`( T, n ) | ( function )

`‣ Bettinumbers`( T ) | ( function )

Inputs a pure cubical complex, or cubical complex, simplicial complex or chain complex T and a non-negative integer n. The rank of the n-th rational homology group H_n(T, Q) is returned. If no value for n is input then the list of Betti numbers in dimensions 0 to Dimension(T) is returned .

Examples: 1 , 2

##### 28.1-4 ChainComplex

`‣ ChainComplex`( T ) | ( function )

Inputs a pure cubical complex, or cubical complex, or simplicial complex T and returns the (often very large) cellular chain complex of T.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12

##### 28.1-5 CechComplexOfPureCubicalComplex

`‣ CechComplexOfPureCubicalComplex`( T ) | ( function )

Inputs a d-dimensional pure cubical complex T and returns a simplicial complex S. The simplicial complex S has one vertex for each d-cube in T, and an n-simplex for each collection of n+1 d-cubes with non-trivial common intersection. The homotopy types of T and S are equal.

Examples: 1 , 2

##### 28.1-6 PureComplexToSimplicialComplex

`‣ PureComplexToSimplicialComplex`( T, k ) | ( function )

Inputs either a d-dimensional pure cubical complex T or a d-dimensional pure permutahedral complex T together with a non-negative integer k. It returns the first k dimensions of a simplicial complex S. The simplicial complex S has one vertex for each d-cell in T, and an n-simplex for each collection of n+1 d-cells with non-trivial common intersection. The homotopy types of T and S are equal.

For a pure cubical complex T this uses a slightly different algorithm to the function CechComplexOfPureCubicalComplex(T) but constructs the same simplicial complex.

Examples: 1

##### 28.1-7 RipsChainComplex

`‣ RipsChainComplex`( G, n ) | ( function )

Inputs a graph G and a non-negative integer n. It returns n+1 terms of a chain complex whose homology is that of the nerve (or Rips complex) of the graph in degrees up to n.

Examples: 1

##### 28.1-8 VectorsToSymmetricMatrix

`‣ VectorsToSymmetricMatrix`( M ) | ( function )

`‣ VectorsToSymmetricMatrix`( M, distance ) | ( function )

Inputs a matrix M of rational numbers and returns a symmetric matrix S whose (i,j) entry is the distance between the i-th row and j-th rows of M where distance is given by the sum of the absolute values of the coordinate differences.

Optionally, a function distance(v,w) can be entered as a second argument. This function has to return a rational number for each pair of rational vectors v,w of length Length(M[1]).

Examples: 1 , 2

##### 28.1-9 EulerCharacteristic

`‣ EulerCharacteristic`( T ) | ( function )

Inputs a pure cubical complex, or cubical complex, or simplicial complex T and returns its Euler characteristic.

Examples:

##### 28.1-10 MaximalSimplicesToSimplicialComplex

`‣ MaximalSimplicesToSimplicialComplex`( L ) | ( function )

Inputs a list L whose entries are lists of vertices representing the maximal simplices of a simplicial complex. The simplicial complex is returned. Here a "vertex" is a GAP object such as an integer or a subgroup.

Examples: 1 , 2 , 3 , 4 , 5

##### 28.1-11 SkeletonOfSimplicialComplex

`‣ SkeletonOfSimplicialComplex`( S, k ) | ( function )

Inputs a simplicial complex S and a positive integer k less than or equal to the dimension of S. It returns the truncated k-dimensional simplicial complex S^k (and leaves S unchanged).

Examples:

##### 28.1-12 GraphOfSimplicialComplex

`‣ GraphOfSimplicialComplex`( S ) | ( function )

Inputs a simplicial complex S and returns the graph of S.

Examples: 1 , 2 , 3

##### 28.1-13 ContractibleSubcomplexOfSimplicialComplex

`‣ ContractibleSubcomplexOfSimplicialComplex`( S ) | ( function )

Inputs a simplicial complex S and returns a (probably maximal) contractible subcomplex of S.

Examples:

##### 28.1-14 PathComponentsOfSimplicialComplex

`‣ PathComponentsOfSimplicialComplex`( S, n ) | ( function )

Inputs a simplicial complex S and a nonnegative integer n. If n=0 the number of path components of S is returned. Otherwise the n-th path component is returned (as a simplicial complex).

Examples:

##### 28.1-15 QuillenComplex

`‣ QuillenComplex`( G ) | ( function )

Inputs a finite group G and returns, as a simplicial complex, the order complex of the poset of non-trivial elementary abelian subgroups of G.

Examples: 1 , 2 , 3 , 4

##### 28.1-16 SymmetricMatrixToIncidenceMatrix

`‣ SymmetricMatrixToIncidenceMatrix`( S, t ) | ( function )

`‣ SymmetricMatrixToIncidenceMatrix`( S, t, d ) | ( function )

Inputs a symmetric integer matrix S and an integer t. It returns the matrix M with M_ij=1 if I_ij is less than t and I_ij=1 otherwise.

An optional integer d can be given as a third argument. In this case the incidence matrix should have roughly at most d entries in each row (corresponding to the d smallest entries in each row of S).

Examples:

##### 28.1-17 IncidenceMatrixToGraph

`‣ IncidenceMatrixToGraph`( M ) | ( function )

Inputs a symmetric 0/1 matrix M. It returns the graph with one vertex for each row of M and an edges between vertices i and j if the (i,j) entry in M equals 1.

Examples:

##### 28.1-18 CayleyGraphOfGroup

`‣ CayleyGraphOfGroup`( G, A ) | ( function )

Inputs a group G and a set A of generators. It returns the Cayley graph.

Examples:

##### 28.1-19 PathComponentsOfGraph

`‣ PathComponentsOfGraph`( G, n ) | ( function )

Inputs a graph G and a nonnegative integer n. If n=0 the number of path components is returned. Otherwise the n-th path component is returned (as a graph).

Examples:

##### 28.1-20 ContractGraph

`‣ ContractGraph`( G ) | ( function )

Inputs a graph G and tries to remove vertices and edges to produce a smaller graph G' such that the indlusion G' → G induces a homotopy equivalence RG → RG' of Rips complexes. If the graph G is modified the function returns true, and otherwise returns false.

Examples: 1

##### 28.1-21 GraphDisplay

`‣ GraphDisplay`( G ) | ( function )

This function uses GraphViz software to display a graph G.

Examples: 1 , 2

##### 28.1-22 SimplicialMap

`‣ SimplicialMap`( K, L, f ) | ( function )

`‣ SimplicialMapNC`( K, L, f ) | ( function )

Inputs simplicial complexes K , L and a function f: K!.vertices → L!.vertices representing a simplicial map. It returns a simplicial map K → L. If f does not happen to represent a simplicial map then SimplicialMap(K,L,f) will return fail; SimplicialMapNC(K,L,f) will not do any check and always return something of the data type "simplicial map".

Examples:

##### 28.1-23 ChainMapOfSimplicialMap

`‣ ChainMapOfSimplicialMap`( f ) | ( function )

Inputs a simplicial map f: K → L and returns the corresponding chain map C_∗(f) : C_∗(K) → C_∗(L) of the simplicial chain complexes..

Examples:

##### 28.1-24 SimplicialNerveOfGraph

`‣ SimplicialNerveOfGraph`( G, d ) | ( function )

Inputs a graph G and returns a d-dimensional simplicial complex K whose 1-skeleton is equal to G. There is a simplicial inclusion K → RG where: (i) the inclusion induces isomorphisms on homotopy groups in dimensions less than d; (ii) the complex RG is the Rips complex (with one n-simplex for each complete subgraph of G on n+1 vertices).

Examples: 1 , 2

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