Notes: | The topics covered include some general topology, smooth manifolds, homology and homotopy groups, duality, cohomology and products. The text is somewhat informal with an occasional offering of two proofs of the same result. Some optional topics are marked with a star, but a good lecturer can use this text to create a fine course at the appropriate level.
There are various innovative things, which are accessible but not traditionally covered. An example would be the Hopf maps, or the calculation of some cohomology of compact simple Lie groups.
The text does lean towards the topology, and some readers might expect more classical geometry (in the style of Frobenius, or the Gauss-Bonnet theorem). One might also quibble about the order in which certain topics occur. For example, the basic notion of a fibre space is more than 150 pages past de Rham's theorem. Thom transversality is more than 100 pages before singular homology. Nevertheless, with a little guidance, a beginning graduate student can use this text to learn a great deal of mathematics. (MathSciNet) |