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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346<Chapter><Heading> Simplicial Complexes</Heading>12<Table Align="|l|" >345<Row>6<Item>7<C>Homology(T,n)</C>8<C>Homology(T)</C>910<P/>11Inputs a pure cubical complex, or cubical complex, or simplicial complex <M>T</M>12and a non-negative integer <M>n</M>. It returns the n-th integral homology of <M>T</M> as a list of torsion integers.13If no value of <M>n</M> is input then the list of all homologies of <M>T</M>14in dimensions 0 to Dimension(T) is returned .15</Item>16</Row>1718<Row>19<Item>20<C>RipsHomology(G,n)</C>21<C>RipsHomology(G,n,p)</C>2223<P/>24Inputs a graph <M>G</M>, a non-negative integer <M>n</M> (and optionally a prime number <M>p</M>). It returns the integral homology (or mod p homology) in degree <M>n</M> of the Rips complex of <M>G</M>.25</Item>26</Row>272829<Row>30<Item>31<Index>Bettinumbers</Index>32<C> Bettinumbers(T,n)</C>33<C> Bettinumbers(T)</C>3435<P/>36Inputs a pure cubical complex, or cubical complex, simplicial complex or chain complex <M>T</M> and a non-negative integer <M>n</M>.37The rank of the n-th rational homology group <M>H_n(T,\mathbb Q)</M> is returned. If no value for n is input then the list of Betti numbers38in dimensions 0 to Dimension(T) is returned .39</Item>40</Row>414243<Row>44<Item>4546<C>ChainComplex(T)</C>4748<P/>49Inputs a pure cubical complex, or cubical complex, or simplicial complex50<M>T</M>51and returns the (often very large) cellular chain complex of <M>T</M>.52</Item>53</Row>5455<Row>56<Item>57<Index>CechComplexOfPureCubicalComplex</Index>58<C>CechComplexOfPureCubicalComplex(T)</C>5960<P/>61Inputs a d-dimensional pure cubical complex62<M>T</M>63and returns a simplicial complex <M>S</M>. The simplicial complex <M>S</M> has one vertex for each d-cube in <M>T</M>, and an n-simplex for each collection of n+1 d-cubes with non-trivial common intersection.64The homotopy types of <M>T</M> and <M>S</M> are equal.65</Item>66</Row>6768<Row>69<Item>70<Index>PureComplexToSimplicialComplex</Index>71<C>PureComplexToSimplicialComplex(T,k)</C>7273<P/>74Inputs either a d-dimensional pure cubical complex75<M>T</M> or a d-dimensional pure permutahedral complex <M>T</M> together with76a non-negative integer <M>k</M>. It77returns the first <M>k</M> dimensions of78a simplicial complex <M>S</M>. The simplicial complex <M>S</M> has one vertex for each d-cell in <M>T</M>, and an n-simplex for each collection of n+1 d-cells with non-trivial common intersection.79The homotopy types of <M>T</M> and <M>S</M> are equal.808182<P/> For a pure cubical complex <M>T</M> this uses a slightly different algorithm to the function CechComplexOfPureCubicalComplex(T) but constructs the same simplicial complex.83</Item>84</Row>85868788<Row>89<Item>90<Index>RipsChainComplex</Index>91<C>RipsChainComplex(G,n)</C>92<P/>93Inputs a graph <M>G</M> and a non-negative integer <M>n</M>. It returns94<M>n+1</M> terms of a chain complex whose homology is that of the nerve (or Rips complex) of95the graph in degrees up to <M>n</M>.96</Item>97</Row>9899<Row>100<Item>101<Index>VectorsToSymmetricMatrix</Index>102<C>VectorsToSymmetricMatrix(M)</C>103<C>VectorsToSymmetricMatrix(M,distance)</C>104105<P/>106Inputs a matrix <M>M</M> of rational numbers and returns a symmetric matrix <M>S</M> whose <M>(i,j)</M> entry is the distance between the <M>i</M>-th row and <M>j</M>-th rows of <M>M</M> where distance is given by the sum of the absolute values of the coordinate differences.107108<P/>109Optionally, a function distance(v,w) can be entered as a second argument. This function has to return a rational number for each pair of rational110vectors <M>v,w</M> of length Length(M[1]).111</Item>112</Row>113114115<Row>116<Item>117<Index>EulerCharacteristic</Index>118<C>EulerCharacteristic(T)</C>119120<P/>121Inputs a pure cubical complex, or cubical complex, or simplicial complex122<M>T</M>123and returns its Euler characteristic.124</Item>125</Row>126127<Row>128<Item>129<Index>MaximalSimplicesToSimplicialComplex</Index>130<C>MaximalSimplicesToSimplicialComplex(L)</C>131132<P/>133Inputs a list L whose entries are lists of vertices representing the maximal simplices of a simplicial complex.134The simplicial complex is returned. Here a "vertex" is a GAP object such as an integer or a subgroup.135</Item>136</Row>137138<Row>139<Item>140<Index>SkeletonOfSimplicialComplex</Index>141<C>SkeletonOfSimplicialComplex(S,k)</C>142143<P/>144Inputs a simplicial complex <M>S</M> and a positive integer <M>k</M> less than or equal to the dimension of <M>S</M>. It returns the truncated <M>k</M>-dimensional simplicial complex <M>S^k</M> (and leaves <M>S</M> unchanged).145</Item>146</Row>147148<Row>149<Item>150<Index>GraphOfSimplicialComplex</Index>151<C>GraphOfSimplicialComplex(S)</C>152153<P/>154Inputs a simplicial complex <M>S</M> and returns the graph155of <M>S</M>.156</Item>157</Row>158159<Row>160<Item>161<Index>ContractibleSubcomplexOfSimplicialComplex</Index>162<C>ContractibleSubcomplexOfSimplicialComplex(S)</C>163164<P/>165Inputs a simplicial complex <M>S</M> and returns a (probably maximal)166contractible subcomplex167of <M>S</M>.168</Item>169</Row>170171<Row>172<Item>173<Index>PathComponentsOfSimplicialComplex</Index>174<C>PathComponentsOfSimplicialComplex(S,n)</C>175176<P/>177Inputs a simplicial complex <M>S</M> and a nonnegative integer <M>n</M>. If <M>n=0</M> the number of path components of <M>S</M> is returned. Otherwise the n-th path component is returned (as a simplicial complex).178</Item>179</Row>180181182183<Row>184<Item>185<Index>QuillenComplex</Index>186<C>QuillenComplex(G)</C>187188<P/>189Inputs a finite group190<M>G</M>191and returns, as a simplicial complex, the order complex of the poset of non-trivial elementary abelian subgroups of <M>G</M>.192</Item>193</Row>194195<Row>196<Item>197<Index>SymmetricMatrixToIncidenceMatrix</Index>198<C>SymmetricMatrixToIncidenceMatrix(S,t)</C>199<C>SymmetricMatrixToIncidenceMatrix(S,t,d)</C>200201<P/>202Inputs a symmetric integer matrix S and an integer t. It203returns the matrix <M>M</M> with <M>M_{ij}=1</M> if <M>I_{ij}</M> is less than <M> t</M>204and <M>I_{ij}=1</M> otherwise.205206<P/>207An optional integer <M>d</M> can be given as a third argument. In this case208the incidence matrix should have roughly at most <M>d</M> entries in each row (corresponding to the $d$ smallest entries in each row of <M>S</M>).209</Item>210</Row>211212<Row>213<Item>214<Index>IncidenceMatrixToGraph</Index>215<C>IncidenceMatrixToGraph(M)</C>216217<P/>218Inputs a symmetric 0/1 matrix M. It219returns the220graph with one vertex for each row of <M>M</M> and an edges between vertices <M>i</M> and <M>j</M> if the <M>(i,j)</M> entry in <M>M</M> equals 1.221</Item>222</Row>223224<Row>225<Item>226<Index>CayleyGraphOfGroup</Index>227<C>CayleyGraphOfGroup(G,A)</C>228229<P/>230Inputs a group <M>G</M> and a set <M>A</M> of generators. It returns the Cayley graph.231</Item>232</Row>233234235<Row>236<Item>237<Index>PathComponentsOfGraph</Index>238<C>PathComponentsOfGraph(G,n)</C>239240<P/>241Inputs a graph <M>G</M> and a nonnegative integer <M>n</M>. If <M>n=0</M> the number of path components is returned. Otherwise the n-th path component is returned (as a graph).242</Item>243</Row>244245<Row>246<Item>247<Index>ContractGraph</Index>248<C>ContractGraph(G)</C>249250<P/>251Inputs a graph <M>G</M> and tries to remove vertices and edges to produce a smaller graph <M>G'</M>252such that the indlusion <M>G' \rightarrow G</M> induces a homotopy equivalence253<M>RG \rightarrow RG'</M> of Rips complexes. If the graph <M>G</M> is modified the function returns true, and otherwise returns false.254</Item>255</Row>256257258<Row>259<Item>260<Index>GraphDisplay</Index>261<C>GraphDisplay(G)</C>262263<P/>264This function uses GraphViz software to display a graph <M>G</M>.265</Item>266</Row>267268<Row>269<Item>270<Index>SimplicialMap</Index>271<Index>SimplicialMapNC</Index>272<C>SimplicialMap(K,L,f)</C>273<C>SimplicialMapNC(K,L,f)</C>274275<P/>276Inputs simplicial complexes <M>K</M> , <M>L</M> and a277function <M>f\colon K!.vertices \rightarrow L!.vertices</M>278representing a simplicial map. It returns a simplicial279map <M>K \rightarrow L</M>. If <M>f</M> does not happen to represent a280simplicial map then SimplicialMap(K,L,f) will return fail; SimplicialMapNC(K,L,f)281will not do any check and always return something of the data type "simplicial map".282</Item>283</Row>284285<Row>286<Item>287<Index>ChainMapOfSimplicialMap</Index>288<C>ChainMapOfSimplicialMap(f)</C>289290<P/>291Inputs a simplicial map <M>f\colon K \rightarrow L</M> and returns the292corresponding chain map <M>C_\ast(f) \colon C_\ast(K) \rightarrow C_\ast(L)</M> of the simplicial chain complexes..293</Item>294</Row>295296297298<Row>299<Item>300<Index>SimplicialNerveOfGraph</Index>301<C>SimplicialNerveOfGraph(G,d)</C>302303<P/>304Inputs a graph <M>G</M> and returns a <M>d</M>-dimensional305simplicial complex <M>K</M> whose 1-skeleton is equal to <M>G</M>. There is a simplicial inclusion <M>K \rightarrow RG</M> where: (i) the inclusion induces isomorphisms on homotopy groups in dimensions less than <M>d</M>; (ii) the complex <M>RG</M> is the Rips complex (with one <M>n</M>-simplex for each complete subgraph of <M>G</M> on <M>n+1</M> vertices).306</Item>307</Row>308309310311312</Table>313</Chapter>314315316317318