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����;� TeX output 2013.10.13:1915�������! systemdict /pdfmark known{userdict /?pdfmark systemdict /exec get put}{userdict /?pdfmark systemdict /pop get put userdict /pdfmark systemdict /cleartomark get put}ifelse�ps:SDict begin [/Producer (dvips + Distiller)/Title (Written with GAPDoc)/Subject ()/Creator (LaTeX with hyperref package / GAPDoc)/Author ()/Keywords () /DOCINFO pdfmark end�o! /DvipsToPDF{72.27 mul Resolution div} def/PDFToDvips{72.27 div Resolution mul} def/BPToDvips{72 div Resolution mul}def/BorderArrayPatch{[exch{dup dup type/integertype eq exch type/realtype eq or{BPToDvips}if}forall]}def/HyperBorder {1 PDFToDvips} def/H.V {pdf@hoff pdf@voff null} def/H.B {/Rect[pdf@llx pdf@lly pdf@urx pdf@ury]} def/H.S {currentpoint HyperBorder add /pdf@lly exch def dup DvipsToPDF 72 add /pdf@hoff exch def HyperBorder sub /pdf@llx exch def} def/H.L {2 sub dup/HyperBasePt exch def PDFToDvips /HyperBaseDvips exch def currentpoint HyperBaseDvips sub /pdf@ury exch def/pdf@urx exch def} def/H.A {H.L currentpoint exch pop vsize 72 sub exch DvipsToPDF HyperBasePt sub sub /pdf@voff exch def} def/H.R {currentpoint HyperBorder sub /pdf@ury exch def HyperBorder add /pdf@urx exch def currentpoint exch pop vsize 72 sub exch DvipsToPDF sub /pdf@voff exch def} def��������@fT��ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end��������color push gray 0�ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Dps:SDict begin [/View [/XYZ H.V]/Dest (page.1) cvn /DEST pdfmark end�	color pop�������LfT���	��color push  Black���\�	color pop��������	����fT�fps:SDict begin [/Count -0/Dest (chapter.1) cvn/Title (Resolutions of the ground ring) /OUT pdfmark end�_ps:SDict begin [/Count -0/Dest (chapter.2) cvn/Title ( Resolutions of modules) /OUT pdfmark end�gps:SDict begin [/Count -0/Dest (chapter.3) cvn/Title ( Induced equivariant chain maps) /OUT pdfmark end�Qps:SDict begin [/Count -0/Dest (chapter.4) cvn/Title ( Functors) /OUT pdfmark end�Xps:SDict begin [/Count -0/Dest (chapter.5) cvn/Title ( Chain complexes) /OUT pdfmark end�_ps:SDict begin [/Count -0/Dest (chapter.6) cvn/Title ( Sparse Chain complexes) /OUT pdfmark end�gps:SDict begin [/Count -0/Dest (chapter.7) cvn/Title ( Homology and cohomology groups) /OUT pdfmark end�Xps:SDict begin [/Count -0/Dest (chapter.8) cvn/Title ( Poincare series) /OUT pdfmark end�bps:SDict begin [/Count -0/Dest (chapter.9) cvn/Title ( Cohomology ring structure) /OUT pdfmark end�ups:SDict begin [/Count -0/Dest (chapter.10) cvn/Title ( Cohomology rings of p-groups \(mainly p=2\)) /OUT pdfmark end�wps:SDict begin [/Count -0/Dest (chapter.11) cvn/Title ( Commutator and nonabelian tensor computations) /OUT pdfmark end�tps:SDict begin [/Count -0/Dest (chapter.12) cvn/Title ( Lie commutators and nonabelian Lie tensors) /OUT pdfmark end�kps:SDict begin [/Count -0/Dest (chapter.13) cvn/Title ( Generators and relators of groups) /OUT pdfmark end�qps:SDict begin [/Count -0/Dest (chapter.14) cvn/Title ( Orbit polytopes and fundamental domains) /OUT pdfmark end�Rps:SDict begin [/Count -0/Dest (chapter.15) cvn/Title ( Cocycles) /OUT pdfmark end�cps:SDict begin [/Count -0/Dest (chapter.16) cvn/Title ( Words in free ZG-modules ) /OUT pdfmark end�Ups:SDict begin [/Count -0/Dest (chapter.17) cvn/Title ( FpG-modules) /OUT pdfmark end�Yps:SDict begin [/Count -0/Dest (chapter.18) cvn/Title ( Meataxe modules) /OUT pdfmark end�Xps:SDict begin [/Count -0/Dest (chapter.19) cvn/Title ( G-Outer Groups) /OUT pdfmark end�Vps:SDict begin [/Count -0/Dest (chapter.20) cvn/Title ( Cat-1-groups) /OUT pdfmark end�[ps:SDict begin [/Count -0/Dest (chapter.21) cvn/Title ( Simplicial groups) /OUT pdfmark end�ops:SDict begin [/Count -0/Dest (chapter.22) cvn/Title ( Coxeter diagrams and graphs of groups) /OUT pdfmark end�]ps:SDict begin [/Count -0/Dest (chapter.23) cvn/Title (Torsion subcomplexes) /OUT pdfmark end�^ps:SDict begin [/Count -0/Dest (chapter.24) cvn/Title ( Simplicial Complexes) /OUT pdfmark end�Zps:SDict begin [/Count -0/Dest (chapter.25) cvn/Title (Cubical Complexes) /OUT pdfmark end�]ps:SDict begin [/Count -0/Dest (chapter.26) cvn/Title (Regular CW-Complexes) /OUT pdfmark end�Yps:SDict begin [/Count -0/Dest (chapter.27) cvn/Title ( Knots and Links) /OUT pdfmark end�|ps:SDict begin [/Count -0/Dest (chapter.28) cvn/Title ( Finite metric spaces and their filtered complexes ) /OUT pdfmark end�vps:SDict begin [/Count -0/Dest (chapter.29) cvn/Title ( Commutative diagrams and abstract categories) /OUT pdfmark end�aps:SDict begin [/Count -0/Dest (chapter.30) cvn/Title ( Arrays and Pseudo lists) /OUT pdfmark end�ops:SDict begin [/Count -0/Dest (chapter.31) cvn/Title ( Parallel Computation - 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������::::::::::::::::::::::::����AutomorphismGroupAsCatOneGroup(G)�����Inputs���a�group��G��and�returns�the�Cat-1-group��C�@��corresponding�to�the�crossed�module��G�n��!��A���ut����(�G�)�.������::::::::::::::::::::::::����HomotopyGroup(C,n)�����Inputs���a�cat-1-group��C�@��and�an�inte���ger�n.�d�It�returns�the��n�th�homotop���y�group�of��C����.������::::::::::::::::::::::::����HomotopyModule(C,2)�����Inputs��|a�cat-1-group��C�L9�and�an��{inte���ger�n=2.��It�returns�the�second�homotop���y�group�of��C�L9�as�a�G-module���(i.e.�d�abelian���G-outer�group)�where�G�is�the�fundamental�group�of�C.������::::::::::::::::::::::::����QuasiIsomorph(C)�����Inputs�@�a�cat-1-group��C��F�and�returns�a�cat-1-group��D��for�which�there�e���xists�some�homomorphism��C���!����D����that���induces�isomorphisms�on�homotop���y�groups.���This���function�w���as�implemented�by��Le�V����an�Luyen�.�������	��color push  Black����65����\�	color pop�����B�p��ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end��������color push gray 0�ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Eps:SDict begin [/View [/XYZ H.V]/Dest (page.66) cvn /DEST pdfmark end�	color pop����������LfT���	��color push  Black����*�66����\�	color pop����������fT�
����	�::::::::::::::::::::::::����	��ModuleAsCatOneGroup(G,alpha,M)�����	��Inputs�]na�group��G�,��an�abelian�group��M���and�a�homomorphism���觿:�7v�G���!���A���ut����(�M��.�)�.�F�It�returns�the����	�Cat-1-group����C�@��corresponding�th�the�zero�crossed�module��0:�7v�M�%�!�n��G�.�������	�::::::::::::::::::::::::����	��MooreComplex(C)�����	��Inputs�T�a�cat-1-group��C��i�and�T�returns�its�Moore�comple���x�as�a�G-comple�x�(i.e.�,�as�a�comple�x�of�groups����	�considered���as�1-outer�groups).�������	�::::::::::::::::::::::::����	��NormalSubgroupAsCatOneGroup(G,N)�����	��Inputs�g"a�g!group��G��with�normal�subgroup��N��D�.�HnIt�returns�the�Cat-1-group��C��޻corresponding�th�the�inclusion����	�crossed���module��N�]0�!�n��G�.�������	�::::::::::::::::::::::::����	��XmodToHAP(C)�����	��Inputs�U&a�U%cat-1-group��C���obtained�from�the�Xmod�package�and�returns�a�cat-1-group��D��for�which�IsHap-����	�CatOneGroup(D)���returns�true.����	�It���returns�"f���ail"�id��C�@��has�not�been�produced�by�the�Xmod�package.��������	��color push  Black���\�	color pop�����C���ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end��������color push gray 0�ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Eps:SDict begin [/View [/XYZ H.V]/Dest (page.67) cvn /DEST pdfmark end�	color pop����������LfT���	��color push  Black���\�	color pop��������	����fT�ps:SDict begin H.S end�ps:SDict begin 13.6 H.A end�Hps:SDict begin [/View [/XYZ H.V]/Dest (chapter.21) cvn /DEST pdfmark end�V��Chapter�/\21��2��color push rgb 0.0 0.0 0.0�Simplicial�8Rgr��eoups�	color pop��-��color push gray 0�����ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Qps:SDict begin [/View [/XYZ H.V]/Dest (L.X7D818E5F80F4CF63) cvn /DEST pdfmark end�	color pop��
������::::::::::::::::::::::::����NerveOfCatOneGroup(G,n)�����Inputs���a�cat-1-group����G��and�a�positi���v���e�inte�ger����n�.��It�returns�the�lo���w-dimensional�part�of�the�nerv���e�of��G����as���a�simplicial�group�of�length��n�.����This�y�function�applies�both�to�y�cat-1-groups�for�which�IsHapCatOneGroup(G)�x�is�true,�h�and�to���cat-1-groups���produced�using�the�Xmod�package.����This���function�w���as�implemented�by��V����an�Luyen�Le�.������::::::::::::::::::::::::����EilenbergMacLaneSimplicialGroup(G,n,dim)�����Inputs�I�a�group�I��G�,���a�positi���v���e�inte�ger��n�,��and�a�positi���v�e�inte�ger�I��d���im�.��The�function�returns�the�rst����1��)+��d���im�B�terms�Aof�a�simplicial�group�with��n��*���)�1�st�homotop���y�group�equal�to��G��and�all�other�homotop���y���groups���equal�to�zero.����This���function�w���as�implemented�by��V����an�Luyen�Le�.������::::::::::::::::::::::::����EilenbergMacLaneSimplicialGroupMap(f,n,dim)�����Inputs�Bua�Bvgroup�homomorphism��f����:�Ge�G��!�Gf�Q�,���a�Bupositi���v���e�inte�ger�Bu�n�,���and�a�positi���v���e�inte�ger��d���im�.���The���function�7�returns�7�the�rst��1���+��d���im��terms�7�of�a�simplicial�group�homomorphism��f�m�:��A�K��<�(�G;�7vn�)��!��K��(�Q;�7vn�)����of���Eilenber�͏g-MacLane�simplicial�groups.�����	��color push  Black����67����\�	color pop�����D����ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end��������color push gray 0�ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Eps:SDict begin [/View [/XYZ H.V]/Dest (page.68) cvn /DEST pdfmark end�	color pop����������LfT���	��color push  Black����*�68����\�	color pop����������fT�
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������Homology(T,n)���Homology(T)�����Inputs�y"a�y!pure�cubical�comple���x,��7or�cubical�comple���x,��7or�simplicial�comple���x��T��Y�and�a�non-ne���g��ati���v�e���inte���ger�X��n�.�9It�returns�X�the�n-th�inte�gral�X�homology�of��T���as�a�list�of�torsion�inte���gers.�9If�no�v���alue�of��n��is���input���then�the�list�of�all�homologies�of��T���in�dimensions�0�to�Dimension(T)�is�returned�.�������RipsHomology(G,n)���RipsHomology(G,n,p)�����Inputs��a�graph���G�,�&�a�non-ne���g��ati���v�e��inte�ger��n��(and��optionally�a�prime�number��p�).�ciIt�returns�the�inte���gral���homology���(or�mod�p�homology)�in�de���gree��n��of�the�Rips�comple�x�of��G�.������::::::::::::::::::::::::����Bettinumbers(T,n)���Bettinumbers(T)�����Inputs���a���pure�cubical�comple���x,�)�or�cubical�comple���x,�)�simplicial�comple�x���or�chain�comple���x��T���and�a���non-ne���g��ati���v�e��-inte�ger��,�n�.��The�rank�of�the�n-th�rational�homology�group��H���z�n����(�T�f�;�7v�Q�)��is�returned.��If�no���v���alue���for�n�is�input�then�the�list�of�Betti�numbers�in�dimensions�0�to�Dimension(T)�is�returned�.�������ChainComplex(T)�����Inputs���a�pure���cubical�comple���x,��or�cubical�comple�x,��or���simplicial�comple�x��T��
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����	�Inputs��a�pure�cubical�comple���x,�K�or�cubical�comple�x,�K�or�simplicial�comple�x��T�2L�and�returns�its�Euler����	�characteristic.�������	�::::::::::::::::::::::::����	��MaximalSimplicesToSimplicialComplex(L)�����	��Inputs�la�list�L�k�whose�lentries�are�lists�of�v���ertices�representing�the�maximal�simplices�of�a�simplicial����	�comple���x.���The��simplicial��comple�x�is��returned.���Here�a�"v���erte�x"�is��a�GAP��object�such�as�an�inte���ger�or�a����	�subgroup.�������	�::::::::::::::::::::::::����	��SkeletonOfSimplicialComplex(S,k)�����	��Inputs� a�¡simplicial�comple���x��S��	�and�a�positi���v���e�inte�ger�¡�k��A�less�than�or�equal�to�the�dimension�of��S�i�.�vyIt����	�returns���the�truncated��k�/��-dimensional�simplicial�comple���x��S��i��y�k��N�(and�lea���v�es��S��5�unchanged).�������	�::::::::::::::::::::::::����	��GraphOfSimplicialComplex(S)�����	��Inputs���a�simplicial�comple���x��S��5�and�returns�the�graph�of��S�i�.�������	�::::::::::::::::::::::::����	��ContractibleSubcomplexOfSimplicialComplex(S)�����	��Inputs���a�simplicial�comple���x��S��5�and�returns�a�(probably�maximal)�contractible�subcomple�x�of��S�i�.�������	�::::::::::::::::::::::::����	��PathComponentsOfSimplicialComplex(S,n)�����	��Inputs�,a�,simplicial�comple���x��S�B��and�a�nonne���g��ati���v�e�,inte�ger��n�.���If�,�n��Ŀ=�0��the�,number�of�path�components����	�of����S��5�is�returned.�d�Otherwise�the�n-th�path�component�is�returned�(as�a�simplicial�comple���x).�������	�::::::::::::::::::::::::����	��QuillenComplex(G)�����	��color push  Black���\�	color pop�����P7��ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end��������color push gray 0�ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Eps:SDict begin [/View [/XYZ H.V]/Dest (page.80) cvn /DEST pdfmark end�	color pop����������LfT���	��color push  Black����*�80����\�	color pop����������fT�
����	�Inputs��}a�nite��~group��G��and�returns,�)as�a�simplicial�comple���x,�)the�order�comple�x�of��~the�poset�of����	�non-tri���vial���elementary�abelian�subgroups�of��G�.�������	�::::::::::::::::::::::::����	��SymmetricMatrixToIncidenceMatrix(S,t)����	�SymmetricMatrixToIncidenceMatrix(S,t,d)�����	��Inputs���a���symmetric�inte���ger�matrix�S���and�an�inte���ger�t.�b�It�returns�the�matrix��M�kʻwith��M���z�ij���P�=�hD1��if��I���z�ij��	��is�less����	�than����t�{e�and��I���z�ij�����=�n�1��otherwise.����	�An���optional���inte���ger��d�g��can�be�gi���v�en���as�a�third�ar�͏gument.��In�this�case�the�incidence�matrix�should�ha���v���e����	�roughly���at�most��d�N��entries�in�each�ro���w�(corresponding�to�the�$d$�smallest�entries�in�each�ro�w�of��S�i�).�������	�::::::::::::::::::::::::����	��IncidenceMatrixToGraph(M)�����	��Inputs��aa��`symmetric�0/1�matrix�M.�It�returns�the�graph�with�one�v���erte�x�for��aeach�ro���w�of��M�P��and�an�edges����	�between���v���ertices��i��and��j��if�the��(�i;�7vj�F�)��entry�in��M�r��equals�1.�������	�::::::::::::::::::::::::����	��CayleyGraphOfGroup(G,A)�����	��Inputs���a�group��G��and�a�set��A��of�generators.�d�It�returns�the�Cayle���y�graph.�������	�::::::::::::::::::::::::����	��PathComponentsOfGraph(G,n)�����	��Inputs�\a�graph��G��and�a�nonne���g��ati���v�e�\inte�ger�\�n�.�B�If��n��l�=��m0��the�number�of�path�components�is�returned.����	�Otherwise���the�n-th�path�component�is�returned�(as�a�graph).�������	�::::::::::::::::::::::::����	��ContractGraph(G)�����	��Inputs�5*a�5)graph��G��and�tries�to�remo���v�e�5*v�ertices�and�5)edges�to�produce�a�smaller�graph��G���y�0���]�such�that�the����	�indlusion�"��G���y�0��Z��!��q�G��induces�a�homotop���y�equi���v�alence�"��RG��!��RG���y�0��ջ�of�Rips�comple���x�es.��0If�"�the�"�graph��G��is����	�modied���the�function�returns�true,�and�otherwise�returns�f���alse.������	��color push  Black���\�	color pop�����Q?���ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end��������color push gray 0�ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Eps:SDict begin [/View [/XYZ H.V]/Dest (page.81) cvn /DEST pdfmark end�	color pop����������LfT���	��color push  Black����*�81����\�	color pop����������fT�
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����	�dimensions���as��M��.�.�d�The�comple���x��R��consist�of�one�cube�selected�at�random�from��M��.�������	�::::::::::::::::::::::::����	��PureCubicalComplexIntersection(S,T)�����	��Inputs��tw���o��pure�cubical�comple���x�es��with�common�dimension�and�array�size.�`�It�returns�the�intersection����	�of��the��tw���o�comple���x�es.��)(An��entry�in�the�binary�array�of�the�intersection�has�v���alue�1�if�and�only�if�the����	�corresponding���entries�in�the�binary�arrays�of�S�and�T�both�ha���v���e�v���alue�1.)�������	�::::::::::::::::::::::::����	��PureCubicalComplexUnion(S,T)�����	��Inputs��2tw���o��1pure�cubical�comple���x�es�with��2common�dimension�and�array�size.�WIt�returns�the�union�of�the����	�tw���o���comple���x�es.�^�(An�entry���in�the�binary�array�of�the�union�has�v���alue�1�if�and�only�if�at�least�one�of�the����	�corresponding���entries�in�the�binary�arrays�of�S�and�T�has�v���alue�1.)�������	�::::::::::::::::::::::::����	��PureCubicalComplexDifference(S,T)�����	��Inputs�!�tw���o�!�pure�cubical�comple���x�es�with�!�common�dimension�and�array�size.��~It�returns�the�dierence����	�S-T��0�.��Q(An��Pentry�in�the�binary�array�of�the�dierence�has�v���alue�1�if�and�only�if�the�corresponding�entry����	�in���the�binary�array�of�S�is�1�and�the�corresponding�entry�in�the�binary�array�of�T�is�0.)�������	�::::::::::::::::::::::::����	��ReadImageAsPureCubicalComplex("file.png",n)�����	��Reads���an���image�le�("le.png",�O�"le.eps",�O�"le.bmp"�etc)�and�an�inte���ger��n��between�0�and�765.����	�It��Lreturns�a�2-dimensional��Kpure�cubical�comple���x�based�on�the�black/white�v���ersion�of�the�image����	�determined���by�the�threshold��n�.�������	�::::::::::::::::::::::::����	��ReadLinkImageAsPureCubicalComplex("file.png")����	�ReadLinkImageAsPureCubicalComplex("file.png",n)�����	��Reads�?�an�image�le�("le.png",���"le.eps",�"le.bmp"�?�etc)�containing�a�knot�or�link�diagram,���and����	�optionally��3a�positi���v���e��4inte�ger��3�n�.�WtThe�inte���ger��n��should�be�a�little�lar�͏ger�than�the�line�thickness�in�the�link�����	��color push  Black���\�	color pop�����TZ���ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end��������color push gray 0�ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Eps:SDict begin [/View [/XYZ H.V]/Dest (page.84) cvn /DEST pdfmark end�	color pop����������LfT���	��color push  Black����*�84����\�	color pop����������fT��	�diagram,��Pand��if�not��pro���vided�then��n��is�set�equal�to�10.��The�function�tries�to�output�the�corresponding��
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and�if�one�of����	�these���checks�f���ails�then�the�function�returns�"f�ail".�������	�::::::::::::::::::::::::����	��ReadImageSequenceAsPureCubicalComplex("directory",n)�����	��Reads���the���name�of�a�directory�containing�a�sequence�of�image�les�(ordered�alphanumerically),����	�and�{xan�{yinte���ger��n��between�0�and�765.��It�returns�a�3-dimensional�pure�cubical�comple���x�based�on�the����	�black/white���v���ersion�of�the�images�determined�by�the�threshold��n�.�������	��Size(T)�����	��This�s�returns�s�the�number�of�non-zero�entries�in�the�binary�array�of�the�cubical�comple���x,��Aor�pure�cubical����	�comple���x���T��0�.�������	��Dimension(T)�����	��This���returns�the�dimension�of�the�cubical�comple���x,�or�pure�cubical�comple�x�T��0�.�������	�::::::::::::::::::::::::����	��WritePureCubicalComplexAsImage(T,"filename","ext")�����	��Inputs��a�2-dimensional�pure��cubical�comple���x�T��0�,�and�a�lename�follo���wed�by�its�e���xtension�(e.g.����	�"myle"���follo���wed�by�"png").�d�A�black/white�image�is�sa���v���ed�to�the�le.�������	�::::::::::::::::::::::::����	��ViewPureCubicalComplex(T)����	�ViewPureCubicalComplex(T,"mozilla")�����	��Inputs�-�a�2-dimensional�pure�cubical�comple���x�T��0�,�-�and�optionally�a�command�such�as�"mozilla"�for����	�vie���wing���image�les.�d�A�black/white�image�is�displayed.��������	��color push  Black���\�	color pop�����Ue���ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end��������color push gray 0�ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Eps:SDict begin [/View [/XYZ H.V]/Dest (page.85) cvn /DEST pdfmark end�	color pop����������LfT���	��color push  Black����*�85����\�	color pop����������fT��	�::::::::::::::::::::::::��
����	��Homology(T,n)����	�Homology(T)�����	��Inputs�y"a�y!pure�cubical�comple���x,��7or�cubical�comple���x,��7or�simplicial�comple���x��T��Y�and�a�non-ne���g��ati���v�e����	�inte���ger�X��n�.�9It�returns�X�the�n-th�inte�gral�X�homology�of��T���as�a�list�of�torsion�inte���gers.�9If�no�v���alue�of��n��is����	�input���then�the�list�of�all�homologies�of��T���in�dimensions�0�to�Dimension(T)�is�returned�.�������	�::::::::::::::::::::::::����	��Bettinumbers(T,n)����	�Bettinumbers(T)�����	��Inputs���a���pure�cubical�comple���x,�)�or�cubical�comple���x,�)�simplicial�comple�x���or�chain�comple���x��T���and�a����	�non-ne���g��ati���v�e��-inte�ger��,�n�.��The�rank�of�the�n-th�rational�homology�group��H���z�n����(�T�f�;�7v�Q�)��is�returned.��If�no����	�v���alue���for�n�is�input�then�the�list�of�Betti�numbers�in�dimensions�0�to�Dimension(T)�is�returned�.�������	�::::::::::::::::::::::::����	��DirectProductOfPureCubicalComplexes(M,N)�����	��Inputs���tw���o�pure�cubical���comple���x�es����M��.;�7vN��"�and�returns�their�direct�product��D��as�a�pure�cubical�comple���x.����	�The���dimension�of��D��is�the�sum�of�the�dimensions�of��M�r��and��N��D�.�������	�::::::::::::::::::::::::����	��SuspensionOfPureCubicalComplex(M)�����	��Inputs�-Ra�pure�cubical�comple���x��M�〻and�returns�a�-Qpure�cubical�comple�x�with�the�homotop���y�type�of�the����	�suspension���of��M��.�.�������	��EulerCharacteristic(T)�����	��Inputs��a�pure�cubical�comple���x,�K�or�cubical�comple�x,�K�or�simplicial�comple�x��T�2L�and�returns�its�Euler����	�characteristic.�������	�::::::::::::::::::::::::����	��PathComponentOfPureCubicalComplex(T,n)�����	��Inputs��ja�pure�cubical�comple���x��T����and�an�inte�ger��n��in��kthe�rane�1,��...,��Bettinumbers(T)[1]�.���It�returns�the�����	��color push  Black���\�	color pop�����Vo���ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end��������color push gray 0�ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Eps:SDict begin [/View [/XYZ H.V]/Dest (page.86) cvn /DEST pdfmark end�	color pop����������LfT���	��color push  Black����*�86����\�	color pop����������fT��	�n-th��-path�component��,of��T�*d�as�a�pure�cubical�comple���x.�The�v���alue��n����=�0��-�is�also�allo���wed,�in�which�case��
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�and���returns�the�(often����	�v���ery���lar�͏ge)�cellular�chain�comple�x�of��T�67�.�������	��ChainComplexOfPair(T,S)�����	��Inputs���a���pure�cubical�comple���x�or�cubical�comple���x��T�Ƹ�and�subcomple�x����S�i�.��It�returns�the�quotient����	��C����(�T�67�)�=C��(�S�i�)��̻of�cellular�chain�comple���x�es.�������	��ExcisedPureCubicalPair(T,S)�����	��Inputs��'a��&pure�cubical�comple���x��T�]�and�subcomple�x��&�S�i�.��It�returns��'the�pair��[�T�!&�n����int���S�i;�7vS�X�n��int�S�i�])��&�of��'pure����	�cubical���comple���x�es���where��int���S����is�the�pure�cubical�comple���x�obtained�from��S����by�remo���ving�its�boundary��I�.�������	�::::::::::::::::::::::::����	��ChainInclusionOfPureCubicalPair(S,T)�����	��Inputs�l*a�l+pure�cubical�comple���x��T��b�and�subcomple�x��S�i�.�sIt�returns�the�l+chain�inclusion��C����(�S��)��Y�!��C����(�T�67�)�l*�of����	�cellular���chain�comple���x�es.�������	�::::::::::::::::::::::::����	��ChainMapOfPureCubicalPairs(M,S,N,T)�����	��Inputs�cza�c{pure�cubical�comple���x��N�Q��and�subcomple�x�es�c{�M��.�,��%�T����and��S�y�in��T�67�.�YIt�returns�the�chain�map����	��C����(�M��.=S�i�)�n��!��C��(�N��D=T�67�)��̻of�quotient�cellular�chain�comple���x�es.�������	�::::::::::::::::::::::::����	��ContractPureCubicalComplex(T)�����	��Inputs���a�pure�cubical�comple���x��T���of�dimension����d�Co�and�remo�v�es��d����-dimensional�cells�from��T���without����	�changing��}the��|homotop���y�type�of��T�67�.��When�the�function�has�been�applied,�)no�further��d����-cells�can�be�����	��color push  Black���\�	color pop�����Wy@��ps:SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end��������color push gray 0�ps:SDict begin H.S end�color push gray 0�	color pop�ps:SDict begin H.R end�Eps:SDict begin [/View [/XYZ H.V]/Dest (page.87) cvn /DEST pdfmark end�	color pop����������LfT���	��color push  Black����*�87����\�	color pop����������fT��	�remo���v�ed���from��T���without�changing�its�homotop���y�type.�d�This�function�modies��T�67�.��
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