The Poincaré Conjecture: In Search of the Shape of the Universe [Hardcover]

I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Unlike John Derbyshire's books on the history of mathematics, this book does not contain much mathematics apart from concepts in topology, but it does contain a good history of the developments leading up to the conjecture in 1904, and the attempts since to provide a proof. The book is intended for a non-specialist audience, but it helps if you have a mathematical background, at least to A level, and are interested in the development of ideas in their historical context. There is not much material on the conjecture itself since to include it would take it out of the scope of a popular book, but one gets the flavour of the idea and its usefulness beyond merely an abtruse mathematical concept. The book is well-written and entertaining and recommended if you don't want a mathematical text-book approach to the subject.