books for the Poincare Conjecture

 The following books have been added to the bibliography.

First I read O’Shea’s “Poincare”; it’s way past time it showed up in the bibliography. Then I bought Weeks and “Ricci” based on O’Shea’s “further reading”. One is high school – barely – and the other is high research. I’ve skimmed Weeks once and have settled down to read it again. “Ricci” is beyond me, as I expected.

While trying to find out how a surface could be given a geometry in which it had constant curvature, I ended up at the back of O’Neill; he’s an old favorite. While trying to sort out topological and differential structures, I took another look at “Instantons” – right up there with “Ricci Flow” – and then discovered why I had bought Bloch in the first place: I could read him. 


Bloch, Ethan D.; A First Course in Geometric Topology and Differential Geometry.
Birkhauser 1997; ISBN 0 8176 3840 7.
[topology, geometry;4 feb 2008]
upper division mathematics.
it is about 3 kinds of structure on surfaces: topological, simplicial, and differential. the restriction to surfaces makes it a quite readable introduction. if you’re like me, you’ve seen each of these structures, but never together.

Freed, Daniel S. and Uhlenbeck, Karen K.; Instantons and Four-Manifolds.
Springer, 1984. ISBN 0 387 96036 8.
[topology, geometry;4 feb 2008]
from the first paragraph of the preface: “This book is the outcome of a seminar…. to go through a proof of Simon Donaldson’s Theorem…. the nonsmoothability of certain topological four-manifolds…. by studying the solution space of … the Yang-Mills equations….”
in other words, this is a highly advanced book. but who could resist the title? certainly not me.

O’Neill, Barrett; Elementary Differential Geometry.
Academic Press, 1997 (2nd Ed); ISBN 0 12 526745 2.
[differential geometry;4 feb 2008]
the first edition revolutionized the teaching of the subject, by introducing differential forms to undergraduates. the prerequisites are multivariate calculus and linear algebra. the 2nd ed. has more material on intrinsic geometry, e.g. more material on the gauss-bonnet theorem. 
this is one of “the books”. it’s also one of the very small set of which i have deliberately bought a later edition primarily to thank the author for the first edition.

O’Shea, Donal; The Poincare’ Conjecture;
Walker Publishing Co., 2007.  OSBN 0 8027 1532 X
[history of math, topology, geometry;4 feb 2008]
“This book is about a single problem. Formulated by a brilliant French mathematician, Henri Poincare, over one hundred years ago….” a pleasant mix of biography and history, and it mentions some fascinating mathematics, but – quite appropriately – it barely scratches the surface. i want more, but this is a good introduction.

Morgan, John and Tian, Gang; Ricci Flow and the Poincare Conjecture.
American Mathematical Society, 2007; ISBN 0 8218 4328 4
[topology;4 feb 2008]
this is the proof of the poincare conjecture. i may never understand it, but i couldn’t resist it. 

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