In Men of Mathematics Eric Temple Bell calls Henri Poincaré “the last universalist”. Universalist or not, he had truly remarkable breadth. He made critical contributions to the modern study of dynamical systems and the qualitative study of differential equations, including the first serious look at chaotic systems. He also worked in complex function theory, integral equations, Diophantine equations and a variety of areas in mathematical physics. The current book provides a skillful translation of his breakthrough work in algebraic topology.
This volume includes Analysis Situs, which Poincaré intended as the primary work, and five supplements. Analysis Situs develops homology theory as a foundation for computing Betti numbers, provides a duality theorem for the Betti numbers, generalizes Euler’s formula to arbitrary dimensions, introduces the fundamental group and asks (without providing an answer) whether the fundamental group alone can distinguish between different three-dimensional manifolds.
The first supplement, Complément a l’analysis situs, revised his homology theory in response to criticism from Heegard. Poincaré may have thought that this Complément would complete his Analysis Situs work, but four more of these were to follow to fix gaps, tighten up loose ends, and pursue related ideas. Much of the fascination to the modern reader comes from watching Poincaré continue to struggle with and refine his ideas. Heegard had pointed out that Poincaré’s definition of Betti numbers was sometimes at odds with his duality theorem. So Poincaré saved the theorem in his Complément by revising his homology theory; among other things, he assumed that all manifolds have a triangulation and made the theory more combinatorial. He concludes the Complément with a proof that every differential manifold has a triangulation. At the end of this proof he says, “One is thus freed of all doubts on the subject.” It is not, however, a very convincing proof. It was thirty-five years before Cairns published the first rigorous proof.
The other four supplements are quite a mix. They include a deeper discussion of homology theory, an investigation of the connectivity of algebraic surfaces, an application of non-Euclidean geometry to surface topology, and a fair amount of meandering. The second supplement describes Poincaré’s discovery of torsion and the first (and incorrect) version of his famous conjecture: The three-dimensional sphere is the only closed three-dimensional manifold with trivial Betti and torsion numbers. Near the very end of the last supplement, Poincaré announces his discovery of what we now call the Poincaré homology sphere. That forces him to revise his conjecture into the form that we know it. He then concludes by suggesting that investigation of the revised conjecture “would carry us too far away.”
John Stillwell has done a marvelous job with his translation and editing. Poincaré’s original text has lots of errors — large and small. Stillwell has corrected typographical and other small errors, but left the serious errors in place and marked them only by footnote. That he left the errors in place is important — they give us the sense of how the subject was developing for Poincaré, how he worked and what his thought processes were like. Stillwell also left in place even the worst of Poincaré’s annoying notation, largely for the same reasons. There are two changes he made throughout to Poincaré’s terminology that are a big help to the modern reader: he changed “variety” to “manifold” (in accordance with current use) and replaced “one-sided” and “two-sided” by “non-orientable” and “orientable” respectively. (Poincaré had created some confusion by also using one- and two-sided to describe separating and non-separating curves in the plane.)
Poincaré’s prose flows smoothly — he was a pretty good writer, after all — and it is a delight to read. Having said that, but there are rough spots where it’s hard to tell what he’s up to, and he does have a tendency to meander. Much of his writing is in the first person. I found this a charming contrast to the impersonality of many modern texts.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.