Writing a book for the lay public on a topic as important, abstract and controversial as the solution of the Poincaré Conjecture is a daunting undertaking. And I applaud Donal O’Shea for being the first to attempt it. Bringing higher level mathematics to the masses in an enjoyable as well as important part of the profession of mathematics in the 21st century. It requires a balancing act between accuracy and accessibility.
O’Shea does an admirable job covering all the mathematics as well as some history and the personal interest side of the story. However, his execution falls short of what one might hope for. His historical portions are limited for the most part to the standard stories found on most math history websites. In addition, the text has numerous typographical and omission errors that impeded my ability to follow the discussion or relax into the book and enjoy it.
In attempting to write for an all-inclusive audience, the author goes from one extreme to the other in discussion the mathematics. In the early chapters he explains every mathematical detail. To the point of even stating how to pronounce "pi". Because he is trying to explain very elementary mathematics yet keep the discussion moving forward, the explanations become convoluted and confusing. These early chapters have the most errors and are very hard to follow.
For me, the tone was set early on with a significant and unnecessary historical error. On page 8, in his discussion of Pythagoras, who was born on the Greek Isle of Samos, the author claims the temple of Hera on Samos was one of the Seven Wonders of the Ancient World. it was not. (I believe the author had the temple of Artemis in mind, which is on mainland Asia Minor, in Ephesus.) Thirty seconds on Google are enough to check something like this.
The middle chapters do attain a good level of explanation. Not surprisingly, these chapters are well written. I found them enjoyable and wished the whole book could have been written in that vein.
The final chapters swing to the other extreme. There is probably no way to explain the actual solution of the Poincaré Conjecture to a non-mathematician accurately. The author seems to have given up on the attempt and written the closing chapters assuming not only a mathematical audience, but one with a substantial background in the relevant fields. He mentions people and theorems with little or no explanation.
As a result, the book feels as if it was written by three different people for three different audiences.
When I first got this book, I hoped I could use it with my college mathematics students, who have voiced an interest in the Millennium Problems. However, I would not recommend this book for students. I found the presentation too hard to follow. In fact, after reading the book I felt little closer to understanding the problem and its solution than before I read the book.
A mathematical audience would probably find the opening chapters pedantic and the latter chapters lacking in details. However, the author does provide numerous endnotes that fill in some of the mathematical details. Though many parts of the book are well written, overall it left me feeling unfulfilled.