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Exact sequence & meaning of a corollary? 
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#1
Apr2107, 11:53 PM

P: 5

on page 4 of the March 17th lecture found at
http://wwwmath.mit.edu/~sara/tolman.lectures/ (you need to scroll down a bit to see the link the March17 lecture) In the first corollary, when they say map 1 = map 4 = 0 do they mean these are 0homomorphisms? Another other dumb question I had is about the bottom corollary on the same page. What is it saying/what does it mean? Where could I find a sketch of it's proof? That might help me better understand the corollary. Or is an accurate sketch of its proof basically be the sequence of theorems & corollaries above it? Thanks a lot :) 


#2
Apr2207, 08:54 PM

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P: 2,340

As as example, it would have been most helpful had you stated that you are reading I probably didn't express my suggestion very gracefully, but it really is good advice and I hope you will take it. Anyway, the answer to your first question is "yes", and this kind of argument is ubiquitious in homological algebra. (Morse theory and equivariant cohomology: great stuff, BTW, and very timely.) It's not clear if you are looking at p. 4 of the lecture on Morse theory, and if so which of the two corollaries at the bottom of that page you mean. Did you notice the notation on p. 7? This should help you in figuring out what the symbols in whatever corollary you are looking at refer to. If that doesn't help, you probably need to study the earlier portions of the course more carefully (or to ask a more specific question here). 


#3
Apr2207, 11:44 PM

P: 5

I hadn't been aware of how to properly post/compose messages in these sorts of forums. So thank you for the advice. I now know.



#4
Apr2307, 01:36 AM

Sci Advisor
P: 2,340

Exact sequence & meaning of a corollary?
Did you try to email Prof. Tolman?



#5
Apr2307, 09:19 PM

P: 5




#6
Apr2407, 10:53 AM

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HW Helper
P: 9,453

the answer to the first question is yes. although i do not see how she deduces all that from that hypothesis.
ok i guess the star at the top means the maps 1 and 4 really are equal, so since 1 =0 then also 4=0. it then follows immediately from the definition of the word exact that 2,3 re as stated. as to the emaning of the corollary, you have to read the notation on oprevious pages. then you see she is trying to calculate the "equivariant cohomology" of the action by S^1. and she is saying how ti find generatiors by looking only at the fixed points. this is a bsic principle of such actions, everythig is eklarned from looking at the fixed points. more precisely she is telling you everything in the equivariant cohomology is coming from the euler class of the bundle E in the following way: each fixed point ahs a certaib height measured by the morse fucntion phi. for each fixed point p, there is an equivariant cohomology class alpha(p), that equals the eukler clas at p but that vanishes at fixed points below p. moreover these generate. so she is decomposing the eukler class into sort of summands, one for each fixed point, and doing so in order of their height on the manifold. does this help? 


#7
May107, 06:45 PM

P: 5

Yes, thank you! It did help :) I finally understand the last few examples now too thanks to your explanations.



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