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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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