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 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:

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?

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:

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?