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The McTutor biography on Poincaré says:

Poincaré's Analysis situs, published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology. For 40 years after Poincaré published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work.

The fundamental group is credited to Poincaré, plus much of homology theory.

What specific problems stimulated his topological work? Was it an outgrowth of his work on celestial mechanics, or automorphic functions, or something else? The McTutor article mentions that he introduced the fundamental group to help classify two-dimensional surfaces, but why was he trying to do that?

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  • $\begingroup$ You may want to read pages 15-35 of Dieudonné's book A History of Algebraic and Differential Topology, 1900-1960. The introduction is dedicated to Poincaré's work on topology. $\endgroup$ – Ben Nov 5 '14 at 8:19
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One motivation, perhaps the principal one, was his work in ordinary differential equations. (And celestial mechanics, as an application of ordinary differential equations). He introduced what is called "qualitative methods" which are based to a large extend on topology. For example, the existence of periodic orbits.

In the introduction to his paper Analysis Situs (1895) Poincare mentions the following motivations: classification of algebraic surfaces, qualitative theory of differential equations, and applications to what is known now as Lie groups.

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  • $\begingroup$ Exactly. He wrote the same in Analyse de ses travaux scientifiques, 1921, p. 101. $\endgroup$ – Ben Nov 5 '14 at 8:14

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