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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 groups</Heading>123<Table Align="|l|" >45<Row>6<Item>7<Index>NerveOfCatOneGroup</Index>8<C>NerveOfCatOneGroup(G,n)</C>91011<P/>12Inputs a cat-1-group <M>G</M> and a positive integer <M>n</M>. It returns the13low-dimensional part of the nerve of <M>G</M> as a simplicial group of length <M>n</M>.1415<Br/>16<Br/>17This function applies both to cat-1-groups for which IsHapCatOneGroup(G) is true, and to cat-1-groups produced using the Xmod package.18<Br/>19<Br/>20This function was implemented by <B>Van Luyen Le</B>.21</Item>22</Row>2324<Row>25<Item>26<Index>EilenbergMacLaneSimplicialGroup</Index>27<C>EilenbergMacLaneSimplicialGroup(G,n,dim)</C>282930<P/>31Inputs a group <M>G</M>, a positive integer <M>n</M>, and a positive integer <M>dim </M>.32The function returns the first <M>1+dim</M> terms of a33simplicial group with <M>n-1</M>st homotopy group equal to <M>G</M> and all other homotopy groups equal to zero.3435<Br/>36<Br/>37This function was implemented by <B>Van Luyen Le</B>.38</Item>39</Row>4041<Row>42<Item>43<Index>EilenbergMacLaneSimplicialGroupMap</Index>44<C>EilenbergMacLaneSimplicialGroupMap(f,n,dim)</C>454647<P/>48Inputs a group homomorphism <M>f:G\rightarrow Q</M>, a positive integer <M>n</M>, and a positive integer <M>dim </M>.49The function returns the first <M>1+dim</M> terms of a50simplicial group homomorphism <M>f:K(G,n) \rightarrow K(Q,n)</M> of Eilenberg-MacLane simplicial groups.5152<Br/>53<Br/>54This function was implemented by <B>Van Luyen Le</B>.55</Item>56</Row>5758<Row>59<Item>60<Index>MooreComplex</Index>61<C>MooreComplex(G)</C>626364<P/>65Inputs a simplicial group <M>G</M> and returns its Moore complex as a <M>G</M>-complex.666768<Br/>69<Br/>70This function was implemented by <B>Van Luyen Le</B>.71</Item>72</Row>7374<Row>75<Item>76<Index>ChainComplexOfSimplicialGroup</Index>77<C>ChainComplexOfSimplicialGroup(G)</C>787980<P/>81Inputs a simplicial group <M>G</M> and returns the cellular chain complex <M>C</M> of a CW-space <M>X</M> represented by the homotopy type of the simplicial group.82Thus the homology groups of <M>C</M> are the integral homology groups of <M>X</M>.838485<Br/>86<Br/>87This function was implemented by <B>Van Luyen Le</B>.88</Item>89</Row>9091<Row>92<Item>93<Index>SimplicialGroupMap</Index>94<C>SimplicialGroupMap(f)</C>959697<P/>98Inputs a homomorphism <M>f:G\rightarrow Q</M> of simplicial groups.99The function returns an induced map100<M>f:C(G) \rightarrow C(Q)</M> of chain complexes whose homology is the integral homology of the simplicial group G and Q respectively.101102<Br/>103<Br/>104This function was implemented by <B>Van Luyen Le</B>.105</Item>106</Row>107108109<Row>110<Item>111<Index>HomotopyGroup</Index>112<C>HomotopyGroup(G,n)</C>113114115<P/>116Inputs a simplicial group <M>G</M> and a positive integer <M>n</M>. The integer <M>n</M> must be less than the length of <M>G</M>. It117returns, as a group, the (n)-th homology group of its Moore complex. Thus HomotopyGroup(G,0) returns the "fundamental group" of <M>G</M>.118119</Item>120</Row>121122123<Row>124<Item>125<Index>Bar Resolution</Index>126<C>Representation of elements in the bar resolution</C>127128129<P/>130For a group G we denote by <M>B_n(G)</M> the free <M>\mathbb ZG</M>-module with basis the lists <M>[g_1 | g_2 | ... | g_n]</M> where the <M>g_i</M> range over <M>G</M>.131132<Br/>133<Br/>134We represent a word135136<Br/>137<Br/>138139<M>w = h_1.[g_{11} | g_{12} | ... | g_{1n}] - h_2.[g_{21} | g_{22} | ... | g_{2n}] + ... + h_k.[g_{k1} | g_{k2} | ... | g_{kn}] </M>140141<Br/>142<Br/>143144in <M>B_n(G)</M> as a list of lists:145146<Br/>147<Br/>148<M> [ [+1,h_1,g_{11} , g_{12} , ... , g_{1n}] , [-1, h_2,g_{21} , g_{22} , ... | g_{2n}] + ... + [+1, h_k,g_{k1} , g_{k2} , ... , g_{kn}] </M>.149150</Item>151</Row>152153<Row>154<Item>155<Index>BarResolutionBoundary</Index>156<C>BarResolutionBoundary(w)</C>157158159<P/>160This function inputs a word <M>w</M> in the bar resolution module161<M>B_n(G)</M>162and returns its image under the boundary homomorphism <M>d_n\colon B_n(G) \rightarrow B_{n-1}(G)</M> in the bar resolution.163164<Br/>165<Br/>166This function was implemented by <B>Van Luyen Le</B>.167</Item>168</Row>169170171<Row>172<Item>173<Index>BarResolutionHomotopy</Index>174<C>BarResolutionHomotopy(w)</C>175176177<P/>178This function inputs a word <M>w</M> in the bar resolution module179<M>B_n(G)</M>180and returns its image under the contracting homotopy181<M>h_n\colon B_n(G) \rightarrow B_{n+1}(G)</M> in the bar resolution.182183<Br/>184<Br/>185This function is currently being implemented by <B>Van Luyen Le</B>.186</Item>187</Row>188189<Row>190<Item>191<Index>Bar Complex</Index>192<C>Representation of elements in the bar complex</C>193194195<P/>196For a group G we denote by <M>BC_n(G)</M> the free abelian group with basis the lists <M>[g_1 | g_2 | ... | g_n]</M> where the <M>g_i</M> range over <M>G</M>.197198<Br/>199<Br/>200We represent a word201202<Br/>203<Br/>204205<M>w = [g_{11} | g_{12} | ... | g_{1n}] - [g_{21} | g_{22} | ... | g_{2n}] + ... + [g_{k1} | g_{k2} | ... | g_{kn}] </M>206207<Br/>208<Br/>209210in <M>BC_n(G)</M> as a list of lists:211212<Br/>213<Br/>214<M> [ [+1,g_{11} , g_{12} , ... , g_{1n}] , [-1, g_{21} , g_{22} , ... | g_{2n}] + ... + [+1, g_{k1} , g_{k2} , ... , g_{kn}] </M>.215216</Item>217</Row>218219<Row>220<Item>221<Index>BarComplexBoundary</Index>222<C>BarComplexBoundary(w)</C>223224225<P/>226This function inputs a word <M>w</M> in the n-th term of the bar complex227<M>BC_n(G)</M>228and returns its image under the boundary homomorphism <M>d_n\colon BC_n(G) \rightarrow BC_{n-1}(G)</M> in the bar complex.229230<Br/>231<Br/>232This function was implemented by <B>Van Luyen Le</B>.233</Item>234</Row>235236<Row>237<Item>238<Index>BarResolutionEquivalence</Index>239<C>BarResolutionEquivalence(R)</C>240241242<P/>243This function inputs a free <M>ZG</M>-resolution <M>R</M>. It244returns a component object HE with components245246<List>247<Item> HE!.phi(n,w) is a function which inputs a non-negative integer <M>n</M> and a word <M>w</M> in <M>B_n(G)</M>. It returns the image of <M>w</M>248in <M>R_n</M> under a chain equivalence <M>\phi\colon B_n(G) \rightarrow R_n</M>.</Item>249<Item> HE!.psi(n,w) is a function which inputs a non-negative integer <M>n</M> and a word <M>w</M> in <M>R_n</M>. It returns the image of <M>w</M> in <M>B_n(G)</M> under a chain equivalence <M>\psi\colon R_n \rightarrow B_n(G)</M>.</Item>250<Item> HE!.equiv(n,w) is a function which inputs a non-negative integer <M>n</M> and a word <M>w</M> in <M>B_n(G)</M>. It returns the image of <M>w</M>251in <M>B_{n+1}(G)</M> under a <M>ZG</M>-equivariant homomorphism252<Br/>253<Br/>254<M>equiv(n,-) \colon B_n(G) \rightarrow B_{n+1}(G)</M>255<Br/>256<Br/>257satisfying258259260<Display>w - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) .261</Display>262263where <M>d(n,-)\colon B_n(G) \rightarrow B_{n-1}(G)</M> is the boundary homomorphism in the bar resolution.264</Item>265</List>266267This function was implemented by <B>Van Luyen Le</B>.268</Item>269</Row>270271<Row>272<Item>273<Index>BarComplexEquivalence</Index>274<C>BarComplexEquivalence(R)</C>275276277<P/>278This function inputs a free <M>ZG</M>-resolution <M>R</M>. It first constructs the chain complex279<M>T=TensorWithIntegerts(R)</M>. The function280returns a component object HE with components281282<List>283<Item> HE!.phi(n,w) is a function which inputs a non-negative integer <M>n</M> and a word <M>w</M> in <M>BC_n(G)</M>. It returns the image of <M>w</M>284in <M>T_n</M> under a chain equivalence <M>\phi\colon BC_n(G) \rightarrow T_n</M>.</Item>285<Item> HE!.psi(n,w) is a function which inputs a non-negative integer <M>n</M> and an element286<M>w</M> in <M>T_n</M>. It returns the image of <M>w</M> in <M>BC_n(G)</M> under a chain equivalence <M>\psi\colon T_n \rightarrow BC_n(G)</M>.</Item>287<Item> HE!.equiv(n,w) is a function which inputs a non-negative integer <M>n</M> and a word <M>w</M> in <M>BC_n(G)</M>. It returns the image of <M>w</M>288in <M>BC_{n+1}(G)</M> under a homomorphism289<Br/>290<Br/>291<M>equiv(n,-) \colon BC_n(G) \rightarrow BC_{n+1}(G)</M>292<Br/>293<Br/>294satisfying295296297<Display>w - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) .298</Display>299300where <M>d(n,-)\colon BC_n(G) \rightarrow BC_{n-1}(G)</M> is the boundary homomorphism in the bar complex.301</Item>302</List>303304This function was implemented by <B>Van Luyen Le</B>.305</Item>306</Row>307308<Row>309<Item>310<Index>Bar Cocomplex</Index>311<C>Representation of elements in the bar cocomplex</C>312313314<P/>315For a group G we denote by <M>BC^n(G)</M> the free abelian group with basis the lists <M>[g_1 | g_2 | ... | g_n]</M> where the <M>g_i</M> range over <M>G</M>.316317<Br/>318<Br/>319We represent a word320321<Br/>322<Br/>323324<M>w = [g_{11} | g_{12} | ... | g_{1n}] - [g_{21} | g_{22} | ... | g_{2n}] + ... + [g_{k1} | g_{k2} | ... | g_{kn}] </M>325326<Br/>327<Br/>328329in <M>BC^n(G)</M> as a list of lists:330331<Br/>332<Br/>333<M> [ [+1,g_{11} , g_{12} , ... , g_{1n}] , [-1, g_{21} , g_{22} , ... | g_{2n}] + ... + [+1, g_{k1} , g_{k2} , ... , g_{kn}] </M>.334335</Item>336</Row>337338<Row>339<Item>340<Index>BarCocomplexCoboundary</Index>341<C>BarCocomplexCoboundary(w)</C>342343344<P/>345This function inputs a word <M>w</M> in the n-th term of the bar cocomplex346<M>BC^n(G)</M>347and returns its image under the coboundary homomorphism <M>d^n\colon BC^n(G) \rightarrow BC^{n+1}(G)</M> in the bar cocomplex.348<Br/>349<Br/>350This function was implemented by <B>Van Luyen Le</B>.351</Item>352</Row>353354355356</Table>357</Chapter>358359360361362