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$$\Lambda$$-spaces and fixed point theorems. (English) Zbl 0182.56904
The paper generalizes Lefschetz fixed point theorems to “$$\Lambda$$-spaces”. In order to define the Lefschetz number $$\Lambda(f)$$ of a map $$f: X\to X$$ we may use any functor $$H_*$$ from the topological category to the category of graded vector spaces and morphisms of degree 0 such that $$H_*$$ satisfies the homotopy axiom, the dimension axiom and agrees with the usual homology on the category of compact polyhedra. Thus $$H_*$$ may be the singular homology or the Čech homology, for example. A map $$f: X\to X$$ is said to be a Lefschetz map if $$\Lambda(f)$$ is defined and if $$\Lambda(f)\ne 0$$ implies that $$f$$ has a fixed point. A space $$X$$ is said to be a $$\Lambda$$-space if every compact map $$f: X\to X$$ is a Lefschetz map. It is shown that if a compact map $$f: X\to X$$ of any space $$X$$ can be factored $$X\stackrel{g}{\rightarrow} Y\stackrel{h}{\rightarrow} X$$ through a $$\Lambda$$-space $$Y$$ such that $$h$$ is compact then $$f$$ is a Lefschetz map; that if a space $$X$$ is dominated, in a certain way, by $$\Lambda$$-spaces then $$X$$ is itself a $$\Lambda$$-space; and that every polyhedron with the Whitehead topology is a $$\Lambda$$-space. It is proved as a corollary that every ANR (for the category of metric spaces) is a $$\Lambda$$-space. This fact has also been proved by A. Granas [Generalizing the Hopf-Lefschetz fixed point theorems for non-compact ANR-s, Proc. Symp. Infinite Dimensional Topology, Baton Rouge 1967, Ann. Math. Stud. 69, 119–130 (1972; Zbl 0235.55008)]. A weaker result along these lines was obtained by F. E. Browder [Trans. Am. Math. Soc. 119, 179–194 (1965; Zbl 0132.18803)].
Reviewer: Jan W. Jaworowski

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects)
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