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.

MSC:

 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