**History of Mathematics**

Volume: 37;
2010;
228 pp;
Softcover

MSC: Primary 01; 55; 57;

Print ISBN: 978-0-8218-5234-7

Product Code: HMATH/37

List Price: $62.00

Individual Member Price: $49.60

**Electronic ISBN: 978-1-4704-1840-3
Product Code: HMATH/37.E**

List Price: $62.00

Individual Member Price: $49.60

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#### Supplemental Materials

# Papers on Topology: Analysis Situs and Its Five Supplements

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*Henri Poincaré*

Translated by John Stillwell

A co-publication of the AMS and the London Mathematical Society

John Stillwell was the recipient of the Chauvenet Prize for Mathematical
Exposition in 2005.

These famous papers, with their characteristic mixture of deep insight and inevitable confusion, are here presented complete and in English for the first time, with a commentary by their translator, John Stillwell, that guides the reader into the heart of the subject. One of the finest works of one of the great mathematicians is now available anew for students and experts alike.

—Jeremy Gray

The AMS and John Stillwell have made an important contribution to the mathematics literature in this translation of Poincaré. For many of us, these great papers on the foundations of topology are given greater clarity in English. Moreover, reading Poincaré here illustrates the ultimate in research by successive approximations (akin to my own way of mathematical thinking).

— Stephen Smale

I am a proud owner of the original complete works in green leather in French bought for a princely sum in Paris around 1975. I have read them extensively, and often during topology lectures I refer to parts of these works. I am happy that there is now the option for my students to read them in English.

—Dennis Sullivan

The papers in this book chronicle Henri Poincaré's journey in
algebraic topology between 1892 and 1904, from his discovery of the
fundamental group to his formulation of the Poincaré
conjecture. For the first time in English translation, one can follow
every step (and occasional stumble) along the way, with the help of
translator John Stillwell's introduction and editorial comments.

Now that the Poincaré conjecture has finally been proved, by
Grigory Perelman, it seems timely to collect the papers that form the
background to this famous conjecture. Poincaré's papers are in
fact the first draft of algebraic topology, introducing its main
subject matter (manifolds) and basic concepts (homotopy and
homology). All mathematicians interested in topology and its history
will enjoy this book.

This volume is one of an informal sequence of works within the
History of Mathematics series. Volumes in this subset,
“Sources”, are classical mathematical works that served as
cornerstones for modern mathematical thought.

#### Table of Contents

# Table of Contents

## Papers on Topology: Analysis Situs and Its Five Supplements

- Cover Cover11
- Title page iii4
- Contents v6
- Translator’s introduction vii8
- Translator’s Bibliography xix20
- On Analysis Situs 122
- Analysis Situs 526
- Supplement to Analysis Situs 7596
- Second supplement to Analysis Situs 111132
- On certain algebraic surfaces: Third supplement to Analysis Situs 135156
- Cycles on algebraic surfaces: Fourth supplement to Analysis Situs 151172
- Fifth supplement to Analysis Situs 179200
- Index 225246
- Back Cover Back Cover1250

#### Readership

Undergraduates, graduate students, and research mathematicians interested in topology and the history of topology.

#### Reviews

The current book provides a skillful translation of his [Poincaré's] breakthrough work in algebraic topology. . . . John Stillwell has done a marvelous job with his translations and editing. . . . Poincaré's prose flows smoothly - he was a pretty good writer, after all - and it is a delight to read. . . . I found this a charming contrast to the impersonality of many modern texts.

-- Bill Satzer, MAA Reviews

The English translation of Poincaré's celebrated work Analysis situs, together with the series of "Compliments” he added between 1895 and 1904, makes available a remarkable resource to the non-francophone community. Darboux wrote of Poincaré that "when he drafted a memoir, he drafted it in one go, limiting himself to some crossings out, without coming back to what he had written”. Stillwell's translation admits us to this pouring forth of Poincaré's ideas, and to their consequences as played out in the supplements. Not only do we get a glimpse of Poincaré at work, but these papers contain the beginnings of the modern notions used in topology today and the connections they make with the ideas of the nineteenth century.

Analysis situs and its supplements have earned their place as the source of many rich ideas that developed into the field of algebraic topology. At the end of the introduction to Analysis situs Poincaré writes, "I do not think then that I have engaged in useless work in writing the present memoir”. The reader more than a century later will find that he or she will not engage in useless work in reading Poincaré's papers. We have Stillwell to thank for making such work easier with his translation.

-- John McCleary, Mathematical Reviews