The Cambridge Colloquium 1916, Vol. 2

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Excerpt from The Cambridge Colloquium 1916, Vol. 2: Analysis SitusThe continuous trans formations and the singularities (in the way of overlapping, etc.) which are allowed in this proof are completely general, so that we are able to avoid the difficulties, foreign to Analysis Situs, which beset those treatments of the subject which restrict attention to analytic transformations or singularities.It will be seen that, aside from this one question which has to be dealt with in order to give significance to the combinatorial treatment, we leave out of consideration all the work that has been done on the point-set problems of Analysis Situs and on its foundation in terms of axioms or definitions other than those actually used in the text. We have also been obliged by lack of space to leave out all reference to the applications. We have not even given a definition of an i-cell by means of a set of equations and inequalities, or the discussion of orientation by means of the signs of determinants. These are to be found in very readable form in Poincare's first paper, where they are given as the basis of his work. They belong properly, however, to the applications of the subject. For in nearly all cases when Poincare (or anyone else) has proved a theorem of Analysis Situs, he has been obliged to set up a machinery which is equivalent to a set of matrices.No attempt has been made to give a complete account of the history and literature of the subject. These are covered for the period up to 1907 by the article on Analysis Situs by Dehn and Heegaard in the Encyklopadie (Vol. IIIi, p. 153), and the more important works subsequent to that date which bear on our part of the subject are referred to in Chap. V. I take pleasure in acknowledging my indebtedness to Professor J. W. Alexander who has read the manuscript and made many valuable suggestions, and also to Dr. Philip Franklin who has helped with the manuscript, the drawings, and the proof-sheets.About the PublisherForgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.comThis book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.