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

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.

### 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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### References:

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