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Once more on the Lefschetz fixed point theorem. (English) Zbl 1122.55002
An endomorphism $$u=(u_q) : E \to E, u_q : E_q \to E_q$$, of a graded vector space $$E=(E_q)$$ is called weakly nilpotent if for each $$q \geq 0$$ and for each $$x \in E_q$$ there is an integer $$n$$ such that $$u^n_q(x)=0$$. If $$u$$ is a weakly nilpotent endomorphism then the Lefschetz number $$\Lambda (u)$$ of $$u$$ is zero. A continuous map $$f:X \to X$$ is a Lefschetz map if the Lefschetz number $$\Lambda (f)$$ of $$f$$ is defined and if $$\Lambda (f) \neq 0$$ implies that $$f$$ has a fixed point. If $$f: (X,X_0) \to (X,X_0)$$ is a map of pairs, then $$f_X : X \to X$$ and $$f_{X_0} :X_0 \to X_0$$ denote the induced maps on the members of the pairs.
The following general version of the Lefschetz fixed point theorem is proved. Theorem. If a continuous map $$f: (X,X_0) \to (X,X_0)$$ is such that $$f_{X_0} :X_0 \to X_0$$ is a Lefschetz map and $$f_* : H(X,X_0) \to H(X,X_0)$$ is weakly nilpotent then $$f_*$$ is a Lefschetz map. Several consequences and applications are given.

##### MSC:
 55M20 Fixed points and coincidences in algebraic topology 55M15 Absolute neighborhood retracts
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