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](http://en.wikipedia.org/wiki/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?



Analogue of Sperner's lemma for Lefschetz theorem?
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. 


