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Jan 4, 2012 at 20:29 comment added John Jiang I recently learned that you can bound the sum of Betti numbers of a real algebraic set using Morse theory. In particular you can get sharp bounds on the number of connected components that way. This is proved by Milnor and Thom. This turned out to be useful for some application of mine in markov chain mixing.
Dec 17, 2010 at 12:02 history answered Kevin H. Lin CC BY-SA 2.5