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**Papers on topology. Analysis Situs and its five supplements. Transl. by John Stillwell.**
*(English)*
Zbl 1204.55002

History of Mathematics (Providence) 37. Providence, RI: American Mathematical Society (AMS); London: London Mathematical Society (LMS). (ISBN 978-0-8218-5234-7/pbk). xx, 228 p. (2010).

As the title reveals, this volume contains the basic papers of Henri Poincaré on algebraic topology: The big article (121 p.) Analysis Situs (1895) and the five Supplements in English translation. In addition there appears a very helpful Translator’s Introduction and many enlightening footnotes. Since Poincaré with his Analysis Situs stands at the beginning (and not like our generation close to the end) of modern algebraic topology, it is not without fascination to recognize how many ideas began more than a century ago. The main points are the invention of the fundamental group, some homology theory and what we call today the Poincaré duality, as well as long discussions of different definitions of the notion of a manifold. These papers contain many formal mistakes, which are the reason for writing many (if not all) supplements. In the first supplement he had to change his definition of Betti numbers, to deal with a counterexample of Heegaard. Betti numbers (and later torsion numbers) are what Poincaré used as homology. For a present reader most interesting is the Poincaré conjecture. Of course he was only interested in dimension 3. His main issue was to find algebraic invariants (like the fundamental group or the Betti numbers) which characterize manifolds. In the beginning he thought that (a version of) this (later called Poincaré conjecture) assertion is trivial, i.e., not needing any proof.

This is what makes the charme of great conjectures.

Finally in the fifth supplement he constructs a 3-dimensional manifold, having the homology of a sphere but with non-trivial fundamental group. This discovery allows him to formulate finally, at the very end of the fifth supplement, a correct formulation of the Poincaré conjecture, ending up with the sentence

“However this question would carry us too far away.”

This sentence is the end of the present volume.

The translator did a very good job in explaining all this to the modern reader.

This is what makes the charme of great conjectures.

Finally in the fifth supplement he constructs a 3-dimensional manifold, having the homology of a sphere but with non-trivial fundamental group. This discovery allows him to formulate finally, at the very end of the fifth supplement, a correct formulation of the Poincaré conjecture, ending up with the sentence

“However this question would carry us too far away.”

This sentence is the end of the present volume.

The translator did a very good job in explaining all this to the modern reader.