On the Lefschetz fixed point theorem. (English) Zbl 1006.55002

Górniewicz, Lech

Math. Slovaca 52, No. 2, 221-233 (2002).

The abstract version of the Lefschetz fixed point theorem is proved which states the following: If \(X\) is a metric space and if \(X_0\subset X\) is such that \(X_0\) absorbs compact sets, \(f\:X\to X\) is a continuous map, \(f(X_0) \subset X_0\) and \(f/X_0\) is a Lefschetz map, then \(f\) is also (on \(X\)) a Lefschetz map. Here a continuous map \(f\:X\to X\) is called a Lefschetz map if the generalized Lefschetz number \(\Lambda (f)\) of \(f\) is well defined and \(\Lambda (f) \neq 0\) implies that the set Fix\((f)\neq 0\). From that theorem three relative versions of the Lefschetz fixed point theorem are deduced which are connected with condensing and \(k\)-set contraction mappings.

Reviewer: Valter Šeda (Bratislava)

MSC:

55M20	Fixed points and coincidences in algebraic topology
54H25	Fixed-point and coincidence theorems (topological aspects)
47H10	Fixed-point theorems
47H11	Degree theory for nonlinear operators

Keywords:

Lefschetz number; fixed point; CAC-map; condensing map; ANR-space

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