The Poincare Conjecture

At a first glance at the Poincaré Conjecture it can be difficult to see what all the fuss is about - it seems to be saying something which is faily obvious. So it's useful to have a book such as The Poincaré Conjecture; : In Search of the Shape of the Universe, in which Donal O'Shea explains what the conjecture is really claiming, and why mathematicians have had such a hard time proving it. The book starts with a look at how people deduced the shape of the earth, pointing out that even after it was circumnavigated, they couldn't be sure what would happen if you kept going north - maybe you would go on for ever, or even reappear in the south.

O'Shea then looks at Euclidean geometry, and explains how non-euclidean alternatives were eventually discovered. He describes the work of Riemann on differential geometry, and then gets on to the work of Poincaré including his rivaly with Klein. This is followed by a chapter on the attempts to prove the conjecture in the 20^th century, culminating in the success of Grigory Perelman at the start of the 21^st

The use of equationsis avoided in this book, but I'm not sure that it's particularly suitable for those without some previous experience of the subject. O'Shea is keen to get across the nature of the work that has been done on the problem, and I would recommend this book to those who know a bit about topology and want to get a glimpse of the more advanced results in the field.