The Lefschetz-Hopf theorem for Q-simplicial maps
In this paper chain complexes and homology groups are taken over \({\mathbb{Q}}\), the rational numbers. If P is a simplicial complex then \(C_{\#}(P)\) denotes the chain complex of P over \({\mathbb{Q}}\) and \(H_*(P)\) the simplicial homology of P over \({\mathbb{Q}}\). Two Lefschetz- type fixed point theorems are formulated. Definition. A continuous function f: \(X\to Y\) is said to be a \({\mathbb{Q}}\)- simplicial map if for each \(\alpha\in \Lambda\), there exists an \(\alpha\) '\(\in Cov(V)\) with \(\alpha'>\alpha\) such that for each \(\beta\in \Lambda\) there exists a chain map \({\bar \phi}\): \(C_{\#}(\bar X_{\alpha '})\to C_{\#}(\bar X_{\beta})\) such that sup(\({\bar \phi}(\bar c))\leq St_{\alpha} f(\sup (\bar c))\) for \(\bar c\in C_{\#}(\bar X_{\alpha'})\). Theorem 2. Let X be a compact Hausdorff space with finitely generated homology. Let \(f: X\to X\) be a \({\mathbb{Q}}\)-simplicial map and let \(\Lambda\) f be the Lefschetz number. If \(\Lambda f\neq 0\), then f has a fixed point. Definition. If X is a compact Hausdorff space and the identity map \(i: X\to X\) is \({\mathbb{Q}}\)-simplicial map, then X is called a \({\mathbb{Q}}\)-simplicial space [\textit{R. Knill}, Ill. J. Math. 14, 40-51 (1970; Zbl 0186.569)]. Theorem 7. Let \(X_ 1,X_ 2,..\). be a nested sequence of metric \({\mathbb{Q}}\)-simplicial spaces. Set \(X=\cap X_ i\). Suppose \(f: X\to X\) is continuous. Without loss of generality it may be assumed that f can be extended to a map \(\hat f: X_ 1\to X_ 1\). Suppose that for \(\epsilon >0\), there exists an m such that for \(k\geq m\) there exists a map \(g_ k: X_ m\to X_ k\) satisfying \(d(g_ k(x),\hat f(x))<\epsilon\), for \(x\in X_ m\). Then f is a \({\mathbb{Q}}\)- simplicial map. If the homology of X is finitely generated and \(\Lambda f\neq 0\), then f has a fixed point.
nested sequence of metric \({\mathbb{Q}}\)-simplicial spaces
scientific article; zbMATH DE number 4086539
scientific article