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another's boundaries applying Approximate to CP Aq-i Aq.i Arbitrary Chains boundaries on KóL boundary F(Cp.Cq Boundary Formula boundary relation Bs-i call the point cells chains and cycles chains Cp chains not meeting concordantly oriented corresponding Cp and Cq CP and F(Cq Cp.Cq of Cp CP.F Cp.F(Cq CP.Xq CPa.Cq*a Cq respectively Cq*a cycles mod Definition deformation chains deformation theo Denote dual complex extension F CP F Cq F Ds-i F(CP F(CPj F(CPl families of cycles geometric intersection hl h2 hr homologous infinite sequence intersection chains intersection Cp intersection Cp.Cq intersection theory K*hi Ka-L l.h.f. about G Lefschetz Levin New York Madeline Levin meeting one another's mesh so small orientable manifold pectively point set common polyhedral chains Principal Theorem prove Theorems regular subdivision sequence of intersection singular chains subcomplexes Topology Us-i mod Xs-i modN