Poincare’s Prize by George Szpiro: a review

The short: I enjoyed this book; I can recommend it to either mathematicians who aren’t topologists but want to understand a little bit about the Poincare Conjecture with a minimum of effort and to the interested lay-person. There is enough here about the history and the characters involved for even an expert to learn something.

Disclaimer: my research area was (is?) knot theory which is a subset of low dimensional topology. I also have met many of the people that the book talks about; in fact one of them, John Thickstun, was on my dissertation committee. I also had a class from one of them (R. H. Bing).

Longer: the book takes you through the start of topology (called analysis situs at first) and traces the origins of the Poincare conjecture. It talks about the various approaches to solve it, what happened with the various approaches and, just as interestingly, what was going on in the lives of those working on the conjecture and its many spin off problems. Along the way you’ll learn (at a superficial, pop-science level) about homotopy groups, homology groups, homology spheres, Heegard surfaces, Dehn’s Lemma (and who ended up proving it!), R. H. Bing’s partial solution, property P for knots, Thurston’s geometrization conjecture, Smale’s work and, yes, Whitehead manifolds!

Of course, the story ends with the introduction of the Ricci flow, Yau’s work and Pearlman’s proof along with the controversy of some of the published proofs of the Poincare conjecture (e. g., the proof that fills in details that got published in a Chinese mathematics journal).

Chapter 7 talks about the conjecture itself. True, the author makes a few subtle errors here and there (e. g., once confusing simple connectivity with being contractible, confusing “one point compactification” with “deformation”, the Poincare space is called the only known homology sphere), but none are serious enough to really confuse the reader.

So, what is the Poincare Conjecture anyway?
I am going to lie simplify just a bit; hopefully someone who has had multi-variable calculus will understand a bit.

First of all, the classical 3-sphere is the set in 4 space. In practice, a topologist says that a space is a three sphere if there is a function that is continuous, one to one, onto and has continuous inverse that takes onto the classical 3-sphere. Such a function is called a homeomorphism.

A 3-manifold is a space that, at every point, locally “looks like” (traditional 3-space); that is, for each point in the manifold, there is a map from the unit “ball” in 3-space () that is a homeomorphism onto a small subset containing the point (e. g., you can parametrize every point in the space by a piece of 3-space). By the way, such a map is called a "chart" (and yes, most people like their manifolds to be second countable).

The n-th homotopy group is the group that is formed by talking all the maps from the n-sphere into the space ; by “group”, I mean the objects that you studied in abstract algebra. The group operation is function composition (we insist that all maps contain a common point called a “base point”), the identity element is the map that takes the n-sphere into a single point. For the group is Abelian (is commutative).

A manifold is said to be closed and compact if the manifold has no boundary and is compact (in the sense of your analysis class OR can be embedded as a closed, bounded subspace of a for some .

The n-dimensional Poincare conjecture (n > 1) says that an n-dimensional manifold that has trivial m-homotopy groups for all not equal to and n-homotopy group infinite cyclic is homeomorphic to the n-sphere.

This was proven a long time ago for n = 2, proven later (1960′s by Stephen Smale) for n greater than or equal to 5. The proof for n = 4 came even later (1980′s; solved by Michael Freedman).

The reviewed book talks about the n=3 case, which Grigori Pearlman solved.

How difficult was this theorem? Smale, Freedman and Pearlman all were awarded Field’s Medals for their work (Pearlman declined his). That is the highest mathematical prize.

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