On a generalization of approximative absolute neighborhood retracts. (English) Zbl 1185.55002

This article generalizes results by R. Skiba and the author [Topology Appl. 156, 697–709 (2009; Zbl 1166.54005)]. A set-valued map \(\phi:X\multimap Y\) between metric spaces is said to be admissible if there exists a metric space \(Z\), a Vietoris map \(p:X\to Z\) and a continuous map \(q:Z\to Y\) such that \(q(p^{-1}(\{x\}))\subset\phi(x)\) for any \(x\in X\). A compactum \(X\) is said to be an AANMR provided that for any \(\varepsilon>0\) there is a locally convex space \(E\), an open set \(U\subset E\), a map \(r:U\to E\) and an admissible map \(\phi:X\multimap U\) such that \(r(\phi(x))\subset B(x;\varepsilon)\) for any \(x\in X\) where \(B(x;\epsilon)\) denotes the \(\varepsilon\)-ball centered at \(x\). The author proves that for an AANMR \(X\) of finite type (with respect to Čech cohomology) each admissible map \(\phi:X\multimap X\) is a Lefschetz map.


55M20 Fixed points and coincidences in algebraic topology
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
47H10 Fixed-point theorems


Zbl 1166.54005