Short version: One can define a version of the Lefschetz fixed point theorem using any homology or cohomology theory. All versions will be true on some topological spaces, since they agree on some topological spaces, but some might be true more generally than others. If two versions have the same generality, one might be more exact, and make stronger statements. Under this criterion of goodness, which one is best?

Potential answer: Is the the theorem true for Cech cohomology on all T3 compact spaces?

Intuition: Suppose there is a continuous map from a topological space X to itself. Does this have a fixed point?

This is an extraordinarily elementary question. It makes sense to ask this question about any topological space. Yet the tools that are typically used to answer it are usually very specific, referring to special topological spaces like $\mathbb R^n$.

The Lefschetz fixed point theorem is a powerful result in this area. It is usually defined in terms of singular homology, which is not an elementary construction at all. But one could easily use another homology or cohomology theory to calculate the matrix chases and, thereby, the number of fixed points.

In particular Cech cohomology seems extraordinarily elementary, being defined entirely in terms of open sets and their intersections. Therefore, one would expect Cech cohomology to be the best one.

Consider, as an example, the topologist's sine curve. All maps from it to itself have a fixed point. But under singular homology it has characteristic 2, not 1. Cech gives characteristic 1 and is thus more exact.

Edit: I'm more interested in the weak than the strong version of the theorem, because the weak version is presumably more generalizable. I'm interested in any sort of information that compares the theorem in different theories even if it doesn't fall into a strict/better worse dichotomy.

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    $\begingroup$ Will -- which Lefschetz fixed point theorem do you have in mind? There is a weak version that says that if the trace is not 0, then there are fixed points. There is a stronger version, which explicitly gives the contribution of each component of the fixed point set. I think it is somewhere in Dold's lectures on algebraic topology. $\endgroup$ – algori Sep 25 '11 at 21:26
  • $\begingroup$ Defining the strong version in higher generality seems hard. I'm more interested in the weak version, but if anyone knows anything about the strong version, I'm listening. $\endgroup$ – Will Sawin Sep 25 '11 at 22:24
  • $\begingroup$ Why do you think there should be a best one? It seems your question is a bit vague, in that your "goodness" relation does not seem to be a partial order. $\endgroup$ – Ryan Budney Sep 25 '11 at 22:55
  • $\begingroup$ I can see why it's not a complete order, but it's hard to see why it's not a partial order. De Rham cohomology, for instance, is not very general at all. If some theories are better in some cases and others are better than others, I would like to know what those cases are. $\endgroup$ – Will Sawin Sep 26 '11 at 0:54
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    $\begingroup$ Will -- I think one can adapt the argument from SGA 5, expos\'e 3 to show that the weak form of the theorem holds for any compact space such that the constant $\mathbb{Q}$-sheaf has a finite injective resolution. Maybe I'll try to write some details of find a reference later. In the mean time here are some refs: Borel et al, Intersection cohomology, Birkh\"auser; Goresky-MacPherson, Local contributions to Lefschetz formula, Inventiones 199?. $\endgroup$ – algori Sep 28 '11 at 15:56

To answer your specific question about compact T3 spaces: First of all, every compact Hausdorff space (T2 space) is automatically a T4 space (a Hausdorff normal space). In the literature one usually says "compact Hausdorff space". A continuum is a compact, connected, metrizable Hausdorff space. There is a famous example of Kinoshita, described in another MO question, of a contractible continuum that has a fixed-point-free self-map. It even embeds in $\mathbb{R}^3$. So, no Lefschetz fixed point theorem in this generality. As that MO question also explains, Borsuk earlier found Cech-acyclic examples.

One possible answer for the natural generality of the Lefschetz fixed point theorem is the version that Lefschetz himself proved: For compact spaces that are ANRs (absolute neighborhood retracts). ANRs are nicely approximable by simplicial complexes in a certain sense, and actually for compact ANRs Cech and singular cohomology always agree. If you accept Lefschetz's level of generality, then the quest to generalize his fixed point theorem beyond singular cohomology looks a bit self-defeating. Still, there is something in the literature of fixed-point theory called a "Lefschetz space" which is simply directly assumed to satisfy some version of his theorem.

I also wouldn't call Čech cohomology "extraordinarily simple". Like many other definitions of (co)homology, it really comes down to constructing a simplicial complex or a simplicial set, or in this case an inverse system of them, and then using simplicial (co)homology. That's not really all that simple. Also using open covers to build simplicial complexes is a brilliant idea, but not a perfect idea. Actually the message of the examples of Kinoshita and Borsuk and others (including one by my mother, that there can't be a Čech Hurewicz theorem) is that there can't be one perfect cohomology theory in topology, not even for compact, connected Hausdorff spaces. Maybe Čech cohomology is still arguably the best, but it isn't even dual (in the sense of universal coefficients) to any reasonable homology theory.

One of the beautiful aspects of de Rham cohomology, even though it is defined only for smooth manifolds and real coefficients (or generalizations of real coefficients), is that it truly doesn't use simplicial complexes or simplicial sets, only calculus. In fact only that partial derivatives commute. Generality is not the virtue that trumps all others.

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    $\begingroup$ Well that pretty clearly answered my question. I guess I'll take a look at those sources and update my intuitions. The way one defines Cech cohomology in etale cohomology, doesn't use simplicial complexes and is a bit more natural, I think. If you carry it over to this context, we're just finding maps between the groups of locally constant functions on the $n$th fibered tensor power of the open cover, viewed as a map of topological spaces. I'm not 100% sure it works though. de Rham cohomology is very elegant if you already have calculus, but you do have to invent calculus to use it. $\endgroup$ – Will Sawin Nov 3 '11 at 21:48
  • $\begingroup$ I always forget that line breaks in comments disappear. Hopefully it's clear where they're supposed to go. $\endgroup$ – Will Sawin Nov 3 '11 at 21:49

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