TITLE: Any map of a contractible complex to itself has a fixed point
QUESTION [6 upvotes]: Reading Lovasz's lecture notes 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 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:

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.)
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?

REPLY [12 votes]: 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 $f$.