Reading [Lovasz's lecture notes](http://arxiv.org/abs/cs/0205031) on evasive graph properties, I encountered the following extension of Brouwer's fixed point theorem:


**Any continues map from a contractible [finite] simplicial complex to itself has a fixed point.**


Lovasz refers this to Lefschetz. Indeed, it seems [Lefschetz fixed-point theorem](http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem) guarantees the existence of a fixed point under some conditions. 
But I could not find a simple proof for Lefschetz theorem, while Brouwer's theorem has an elementary proof using Sperner's Lemma.


My questions are:


1. How should one interpret the conditions of Lefschetz fixed-point theorem,
and why do they hold for a contractible complex?
(A good reference will be appreciated.)
2. Is there an easy way to prove the above
statement, for the restricted case
of a collapsible complex, and a
simplicial, bijective map,
without using homology?



Any map of a contractible complex to itself has a fixed point
You have to assume that your complex is finite. Then the Lefschetz Fixed Point Theorem definitely says that $f$ must have a fixed point if the (homologically defined) Lefschetz number of $f$ is not zero. If the complex is contractible, then the Lefschetz number of $f$ must be $1$.


This fact about contractible complexes reduces to the Brouwer Theorem if you use the following fact: any finite contractible complex is a retract of $D^n$ for some $n$. That is, there exist continuous maps $i:X\to D^n$ and $r: D^n\to X$ such that $r\circ i$ is the identity. For continuous $f:X\to X$, any fixed point of $i\circ f\circ r:D^n\to D^n$ gives you a fixed point of $f$.


