Customer Reviews

Three-Dimensional Geometry and Topology, Vol. 1

5 star:		(2)
4 star:		(1)
3 star:		(1)
2 star:		(0)
1 star:		(0)

 

 

A must for anyone entering the field of three-dimensional topology and geometry. Most of it is about hyperbolic geometry, which is the biggest area of research in 3-d geometry and topology nowdays.

Most of it is readable to undergraduates. Its target audience, though, is beginning graduate students in mathematics. If not already familiar with hyperbolic geometry, you might want to get an introduction to the subject first. Once with this background, though, you will discover there is another level of understanding of hyperbolic space you never realized was possible. One imagines Thurston able to skateboard around hyperbolic space with the kind of geometric understanding he conveys here.

What made Thurston so famous and successful as a pioneer in 3-d topology and geometry was his other-worldly geometric intuition. This book takes the reader along the first step of the 10000 miles of getting to that intuition.

FAR worse than Thurston's classic notes, whence it came. Covers very little material by comparison, and has a lot more bogus "explanations". Given that the notes are available for free from Math. Sciences Research Inst. (MSRI), in convenient PDF form (for us Kindle DX Wireless Reading Device (9.7" Display, Global Wireless, Latest Generation)and Apple iPad MB292LL/A Tablet (16GB, Wifi) users, this is not a great book to get.

Stanislaw Ulam once compared learning mathematics to learning a language, in that some people learn mathematics by "grammar" while other learn it by ear. Thurston's book is a bit like learning by ear.

I have this book by Jeffery Weeks that Thurston's opens up with the real mathematics: The Shape of Space (Pure and Applied Mathematics). In both cases though, they fail to make the connection of the higher Cartan Lie Algebras to 3 manifold theory clear. When the connection of string theory to 3 dimensional geometry comes through this
approach to geometry, you think that we have a theory that neglects 5 dimensions and above?
So as good as this text is, it still falls somewhat short of what is needed in the modern world and we are forced to think of our own way to interpret the standard model symmetry breaking of SU(5) (Cartan A_4) to U(1)*SU(2)*SU(3).