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Sorry if this is easy/well-known, I don't know much algebraic topology and I'm just curious about this question.

One of the easier proofs of the Brouwer fixed-point theorem (we'll say for n = 2 for concreteness in the terminology, although it generalizes) proceeds by establishing Sperner's lemma and noting that a continuous map gives us a Sperner labeling on a triangulation of our disk. All the "real work" in this proof is in establishing Sperner's lemma, which can be done completely combinatorially.

So I know that the Lefschetz fixed-point theorem generalizes Brouwer's theorem, and that it applies to more general spaces than Brouwer. Is there a (relatively) simple combinatorial statement, analogous to Sperner, that can be easily shown to imply the Lefschetz theorem, at least on some large class of topological spaces?

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    $\begingroup$ You might want to have a look at a recent paper of Nikolai Ivanov's, on the arXiv. He shows that the Sperner Lemma proof of Brouwer's fixed-point theorem is in some sense Brouwer's proof, but put into the simplicial cohomology setting, at the cochain level. Perhaps you could try reverse-engineering Ivanov's paper to construct a simplicial cochain level proof of Lefschetz. $\endgroup$
    – Ryan Budney
    Nov 8, 2009 at 3:55
  • $\begingroup$ Like I said, I don't know a lot of algebraic topology, but it might be worth checking out. Do you have a link? (Ivanov's a frustratingly common name.) $\endgroup$
    – Harrison Brown
    Nov 8, 2009 at 4:59
  • $\begingroup$ Here you go: front.math.ucdavis.edu/0906.5193 Do you have a particular application in mind, or are you just interested if there's a broader analogy between fixed point theorems and combinatorics? IMO you'll probably find the Ivanov paper informative on that level. $\endgroup$
    – Ryan Budney
    Nov 8, 2009 at 5:32
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    $\begingroup$ BTW, while it is true that Lefschetz theorem implies Brouwer's theorem this implication relies on homology theory that Sperner's lemma avoids. Once the homology machinary is at hand, in some sense, Lefschetz theorem is a simple linear algebra fact about chain complexes for which we can expect "easier" combinatorial analogs than sperner's lemma. $\endgroup$
    – Gil Kalai
    Nov 8, 2009 at 16:03

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Probably there aren't any such combinatorial statements in the literature, the difficulty being that the statement of Lefschetz theorem involves looking at the induced action on the homology.

There is a combinatorial lemma - Tucker lemma - which implies the Borsuk-Ulam theorem. Both the Sperner Lemma and Tucker lemma had many generalizations (Ky Fan contributed many, there is a famous deep extension of Sperner lemma by Shapley), although there is less activity in this line of reaserch these days.

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