exact sequence & meaning of a corollary?

Firefly!

on page 4 of the March 17th lecture found at
http://www-math.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 0-homomorphisms?

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 :)

Chris Hillman

Can I offer a bit of advice about asking questions like this? I think you should get in the habit of attributing a writing to an author by name, not by pronoun ("they"). Like bad grammar or bad spelling, this can make a poor impression on at least some of the scholarly minded PF members who are best able to help you. More to the point, there are good reasons for the scholarly convention of citing your sources carefully, and these reasons apply to questions posted in forums like PF, just as they would in a email to a faculty member at a university, or in a research paper. Ultimately it is in your own best interest, as well as that of lurkers who might have similar questions, to observe such basic scholarly conventions to the best of your ability. IOW, when in Rome, do as the romans do. Or in still other words, the point of the motto of William F. Friedman (quoted in the title of my reply) is that taking the time to think things through usually pays off in the end.

As as example, it would have been most helpful had you stated that you are reading

As you see, I googled to find her home page in order to verify that she is on the faculty at UIUC, as stated in the title page of the lecture notes, not at MIT, which is where you found these notes. Formulating your question like this would encourage us to take it more seriously because it would show you had made some effort in composing the post.

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.)

Er.. so should we assume that you are not a student in the course in question? Because if so--- to state the obvious--- you should really ask Prof. Tolman!

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).

Firefly!

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.

No I am not a student of Prof. Tolman. I found her lecture notes online, the notes say they are from 2002.

I meant the very last corollary on page 4. "Assume all the fixed points ...". I had noticed the last page had a summary of the notation so I am aware of what all the symbols mean. To attempt to make my question more specific, is there a geometrical or simplified description/statement of what this corollary is saying? I am having difficulty seeing the big picture w.r.t. this corollary, despite already having studied the earlier portions.

Chris Hillman

Did you try to email Prof. Tolman?

Firefly!

No I did not. I do not know Prof. Tolman, nor do I go to UIUC. I know that is no excuse. I merely thought I would first try to find somebody that might be able to help me understand this math on a forum like this one.

mathwonk

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?

Firefly!

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