### 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 .

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##### 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.

##### 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 .

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##### 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.

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##### 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.

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##### 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.

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##### 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.

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##### 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).

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##### 28.1-9 EulerCharacteristic
 ‣ EulerCharacteristic( T ) ( function )

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

##### 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.

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##### 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).

##### 28.1-12 GraphOfSimplicialComplex
 ‣ GraphOfSimplicialComplex( S ) ( function )

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

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##### 28.1-13 ContractibleSubcomplexOfSimplicialComplex
 ‣ ContractibleSubcomplexOfSimplicialComplex( S ) ( function )

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

##### 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).

##### 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.

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##### 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).

##### 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.

##### 28.1-18 CayleyGraphOfGroup
 ‣ CayleyGraphOfGroup( G, A ) ( function )

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

##### 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).

##### 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.

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##### 28.1-21 GraphDisplay
 ‣ GraphDisplay( G ) ( function )

This function uses GraphViz software to display a graph G.

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##### 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".

##### 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..

##### 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).

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