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.


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