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I think the Lefschetz trace formula says something like:

if $F: X \to X$ is a continuous map of compact manifolds, then

$\chi(X^F) = \sum (-1)^i \mathrm{Tr} f_*|_{H_i(X)}$

First of all, this statement is not quite right even when the fixed points $X^F$ are isolated, I am supposed to somehow put in the indices, right?

So what are the indices and how do I put them in when the fixed points are not isolated?

But, also:

Under what hypotheses can I use it, or something similar, for maps between non-compact, infinite dimensional, etc., etc., spaces?

or perhaps I am asking

What is the most general version of the Lefschetz trace formula?

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By \chi(X^F) do you mean the Euler char? And by "indices" you mean multiplicities and sign? If we want H_i(X) we should assume X is orientable in order for the constant sheaf to be the orientation sheaf (think of RP^2 and a matrix in PGL_3(R) that has real eigenvalues). If not isolated, I don't think putting indices would work; f may be homotopic to another map that has isolated fixed pts. We may want the intersection number in X*X between the diagonal and the graph. Think of f=id, which gives Gauss-Bonnet. In his early version of the thm, Lefschetz worked with closed manifolds.... –  shenghao Dec 16 '10 at 12:27
    
...and then he generalized the thm several times. Requiring it to be a finite CW complex will be ok. I'm not sure about the non-compact or infinite dim cases in topology (there might be no finite triangulation on the space...). In alg geom, when considering varieties over a finite field and f=Frobenius, it works for non-proper schemes (as long as it's of finite type; one uses compact support l-adic cohomology instead of homology) and certain "infinite dim varieties", by which I mean algebraic stacks of finite type. –  shenghao Dec 16 '10 at 12:36
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