History of Topology by I. M. James

By I. M. James

Topology, for a few years, has been probably the most fascinating and influential fields of study in sleek arithmetic. even supposing its origins should be traced again a number of hundred years, it used to be Poincaré who "gave topology wings" in a vintage sequence of articles released round the flip of the century. whereas the sooner historical past, also known as the prehistory, can be thought of, this quantity is principally fascinated by the more moderen background of topology, from Poincaré onwards.
As might be obvious from the checklist of contents the articles conceal quite a lot of issues. a few are extra technical than others, however the reader with no good deal of technical wisdom may still locate many of the articles obtainable. a few are written via specialist historians of arithmetic, others by way of historically-minded mathematicians, who are inclined to have a unique perspective.

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Poincare (1900) presented a new definition and a calculus for the calculation of Betti numbers and torsion from the incidence matrices E^^^ of a cell decomposition of M. The method used diagonalization of incidence matrices by elementary transformations to matrices T^^\ Expressed in sHghtly more structural terms Poincare developed a calculus to choose generators of the Z-module Cq of cellular ^-chains such that all boundary operators dq \Cq -^ Cq-\ are diagonahzed. That allowed him to read off immediately the Betti numbers and torsion coefficients and the distinction between manifolds "with" or "without" torsion from the diagonahzed matrices 7^^^ (Poincare, 1900, p.

2 (30 In the fifth complement (1904) Poincare introduced even more construction procedures, a "skeleton" representation of 3-dimensional manifolds, containing some ideas of three-dimensional Morse theory (Poincare, 1904, pp. ), and an adaptation of an idea of P. Heegard to form a closed 3-manifold M by boundary identification of two homeomorphic handle bodies V and V'P He used these procedures ^^ Poincare would not read M as literal union of the Vi. In lower dimensional components of intersection Ui Pi Uj he thought in terms of a disjoint union, only in /i-dimensional components he would identify the respective points of Ui and Uj; therefore he did not treat M globally as subset of R'".

28 Cf. [Lutzen, 1988, 1995]. The concept of manifold, 1850-1950 35 Most important among the geometrization arguments in this problem field were the following: (1) The subsumption of the least action principle for conservative systems under the form of a geodetical line. The state space was endowed with a physical metric of the form d^^ == 2(H — V) YlSij ^^i ^Qj^ ^^^ ^i coordinates in the state space, gij the metric induced on state space by the metric of the geometric coordinate space, H total energy, and V potential energy.

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