Papers on Topology: Analysis Situs and Its Five Supplements (History of Mathematics)
Language: English
Pages: 228
ISBN: 0821852345
Format: PDF / Kindle (mobi) / ePub
John Stillwell was the recipient of the Chauvenet Prize for Mathematical Exposition in 2005. The papers in this book chronicle Henri Poincaré's journey in algebraic topology between 1892 and 1904, from his discovery of the fundamental group to his formulation of the Poincaré conjecture. For the first time in English translation, one can follow every step (and occasional stumble) along the way, with the help of translator John Stillwell's introduction and editorial comments. Now that the Poincaré conjecture has finally been proved, by Grigory Perelman, it seems timely to collect the papers that form the background to this famous conjecture. Poincaré's papers are in fact the first draft of algebraic topology, introducing its main subject matter (manifolds) and basic concepts (homotopy and homology). All mathematicians interested in topology and its history will enjoy this book. This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, "Sources", are classical mathematical works that served as cornerstones for modern mathematical thought.
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groups corresponding to Riemann surfaces with the same connectivity are isomorphic. Moreover, it is evident that the fuchsian group is none other than the fundamental group g of the surface R, considered as a two-dimensional manifold. We remark that not all fuchsian groups arise in this way from a closed twodimensional manifold. Consider the fuchsian fundamental polygon R0 and, if the fuchsian function exists over the whole plane, it is necessary to adjoin its mirror image R0 in the real axis;
72 Analysis Situs S= α γ β δ be a linear substitution with integer coefficients such that αδ − βγ = 1. Let T = ω1 ω1 ω2 ω2 be another linear substitution with integer coefficients such that ω1 ω2 − ω1 ω2 = 1. The substitution T −1 ST , which is called the transform12 of S by T , is also linear with integer coefficients and of determinant 1. If two linear substitutions S and S with integer coefficients and determinant 1 are transformed into each other by a substitution T , I say that S and
of the first class §17. The case where p is odd 93 γq+2 regions vq+2 of the first class ····················· γp regions vp of the first class. In suppressing the regions of the first class with at least p dimensions and uniting the γp regions vp which make up wp we diminish αp by γp − 1, αp−1 by γp−1 , . . . , αq+1 by γq+1 , αq by 1. Therefore, by virtue of equation (A), the number N will not change. Next we suppress all the manifolds of the second class with at least p − 1 dimensions, in
1894) I published a memoir entitled Analysis situs, or the study of manifolds in spaces of more than three dimensions and the properties of the Betti numbers. Since I shall have occasion to mention this memoir frequently in what follows, I shall use simply the title Analysis situs. The following theorem is found in that memoir: For any closed manifold, the Betti numbers equally distant from the extremes are equal. The same theorem was announced by M. Picard in his Th´eorie des fonctions
The n numbers (3) Mn−1 , Mn−2 , Mn−1 Mn−3 , Mn−2 ··· , M1 , M2 M0 M1 may be called the invariants of the table T . We may remark: 10 Each of these invariants divides its successor; 20 Any of these invariants can be zero, but if one of them is, so are all its successors. If the table T has more rows than columns the reduction is made in the same manner, except that the rˆoles of rows and columns are interchanged. We then have m < n; the number M0 will be the greatest common divisor of the
