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0-cells and 1-cells 1-cell joining 2-cells of C2 Analysis Situs boundary called cells of Cn Chap circuits closed curve closed manifold coefficients of torsion columns of Dk complete set complex Cn complex composed composed of cells defined definition deformation denoted determined dimensional complex elements equations Euclidean fc-cells fc-circuits finite number follows fundamental group Hence homeomorphic homology i-cells incidence relations integers invariant factors J. W. Alexander l)-cells l)-circuits linear combination linear graph linear relation linearly dependent main diagonal negatively related non-singular normal form number of 0-cells obtained one-dimensional complex operations of type orientable manifold oriented 1-cells oriented 1-circuit oriented complex plex Poincare point in common point of Cn positively related proved regular subdivision rows and columns rth row set of 1-circuits set of oriented set of points simplex solutions sphere square matrix straight 1-cells sum mod theorem transformations triangle star two-dimensional complex two-dimensional manifold vertices zero