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another's boundaries Aºi Approximate to Cr ARBITRARY CHAINs boundaries on K-L boundary relation call the point Caº cells chains and cycles chains Cp chains not meeting chains of F concordantly oriented Cºº corresponding Cp and Ca cycles mod cycles mod N cycles on K-L Definition deformation chains deformation theo Denote distance on K—N dual complex F C4 F Cº F Cp F(Ca F(Cº F(Cp F(Dº families of cycles geometric intersection h hr h. f. on K-L h9 hj hr homologous infinite sequence intersection chains intersection theory j=1 r r K*-neighborhood KP-L L. T. chap manifold meeting one another's mesh of K mesh of K7 mesh so small mod L modN point set common polyhedral chains Principal Theorem prove Theorems respectively sequence of intersection singular chains subcomplexes Topology Us-1 mod X-1 mod