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A History of Algebraic and Differential Topology, 1900 - 1960

Part of the Modern Birkhäuser Classics Book Series

DOI: 10.1007/978-0-8176-4907-4

ISBN: 978-0-8176-4906-7

Publisher: Birkhäuser Boston - Springer Journals

May 29, 2009

1 - 10 of 21 Chapters

Available

Introduction

Dieudonné, Jean

Pages: 3 –14

[Ever since the concept of homeomorphism was clearly defined, the “ultimate” problem in topology has been to classify topological spaces “up to homeomorphism”. That this was a hopeless undertaking was very soon apparent, the subspaces of the plane R2 being an obvious example. From this...
Available

The Work of Poincaré

Dieudonné, Jean

Pages: 15 –35

[Concepts and results belonging to algebraic and differential topology may already be noted in the eighteenth and nineteenth centuries, but cannot be said to constitute a mathematical discipline in the usual sense. Before Poincaré we should therefore only speak of the prehistory of algebraic...
Available

The Build-Up of “Classical” Homology

Dieudonné, Jean

Pages: 36 –59

[It took about 30 years to construct a theory of homology applicable to curvilinear “polyhedra,” embodying all the ideas of Poincaré and entirely rigorous. In a period in which the number of professional mathematicians was definitely on the increase, it is surprising that this new field of...
Available

The Beginnings of Differential Topology

Dieudonné, Jean

Pages: 60 –66

[We have seen (chap. I, §2) that Poincaré already had a conception of n-dimensional manifolds of class Cr (r integer ≥ 1 or ∞) and of analytic manifolds (sometimes called Cw manifolds) which was essentially the modern one.]
Available

The Various Homology and Cohomology Theories

Dieudonné, Jean

Pages: 67 –157

[After 1925 topologists began to look for an extension of the notion of homology groups, until then restricted to the very special and rather artificial case of cell complexes, that would have meaning for the most general topological spaces possible; several such extensions were proposed. The...
Available

Introduction

Dieudonné, Jean

Pages: 161 –166

[I have gathered in Part 2 various results to homology which do not use homotopy theory; most of them were discovered prior to its creation. I think it is better do dissociate them form the foundational material described in Part 1, where they would have disgressed form the main themes.]
Available

The Concept of Degree

Dieudonné, Jean

Pages: 167 –181

[L.E.J. Brouwer (1881-1966) had started his career with papers on geometry and mechanics, but by 1909 he had shifted his interests to parts of mathematics far less popular at that time, and in which he was entirely self-taught. He first tackled Hilbert’s famous “5th problem” and showed that all...
Available

Dimension Theory and Separation Theorems

Dieudonné, Jean

Pages: 182 –196

[Before 1870 mathematicians only dealt with those subsets X of an RN that could (at least locally) be “parametrized” by (usually C1) injective maps into X of open subsets of some Rn. It was tacitly assumed that the position of a point in Rn could only be completely determined by a system of n...
Available

Fixed Points

Dieudonné, Jean

Pages: 197 –203

[Brouwer had been considering continuous maps of the sphere S2 into itself as early as 1909; he first studied the particular case of a bijection* f (which is therefore bicontinuous) preserving orientation, and he gave a proof that in that case there exists at least one fixed point x for f, i.e.,...
Available

Local Homological Properties

Dieudonné, Jean

Pages: 204 –213

[Local properties of topological spaces were considered at the beginning of the twentieth century, chiefly by Schoenflies, who was a pioneer in that matter. They were mainly studied for subsets of R2, and without any intervention of homological notions. Examples of these properties are...

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