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 \(H_n(T,\mathbb Q)\) 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

##### 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 \(G' \rightarrow G\) induces a homotopy equivalence \(RG \rightarrow RG'\) 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 \(f\colon K!.vertices \rightarrow L!.vertices\) representing a simplicial map. It returns a simplicial map \(K \rightarrow L\). 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 \(f\colon K \rightarrow L\) and returns the corresponding chain map \(C_\ast(f) \colon C_\ast(K) \rightarrow C_\ast(L)\) 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 \(K \rightarrow RG\) 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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