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The shape genus of a shape map. (English) Zbl 0555.55011
Let \(f: X\to Y\) and \(g: Z\to Y\) be maps of spaces and \({\mathfrak F}\) be a class of functors from the homotopy category of topological spaces into any other category. The author defines the \({\mathfrak F}\)-genus of (f,g) to be the least integer \(k\geq 1\) for which there are open sets \(V_ m\) and maps \(h_ m: V_ m\to X\), \(1\leq m\leq k\), such that \(Z=\cup V_ m\) and \(f\circ h_ m\) and \(g\circ j_ m\) are \({\mathfrak F}\)-equal, where \(j_ m: V_ m\to Z\) is inclusion (1\(\leq m\leq k)\); if no such integer exists, then the \({\mathfrak F}\)-genus is \(\infty\). Being \({\mathfrak F}\)-equal means that \(F([f\circ h_ m)]=F([g\circ j_ m)]\) for all \(F\in {\mathfrak F}\). This concept of \({\mathfrak F}\)-genus is said to generalize that of the genus of a map \(f: X\to Y\) as in [I. Berstein and T. Ganea, Fundam. Math. 50, 265-279 (1962; Zbl 0192.293)].
An investigation is made of the dependency of \({\mathfrak F}\)-genus on \({\mathfrak F},f,g\). Then an extension is made so that \({\mathfrak F}\)-genus can be defined when the maps are maps of inverse systems. This leads to a definition of shape \({\mathfrak F}\)-genus for any pair \(f: X\to Y\) and \(g: Z\to Y\) of shape maps of spaces by taking it to be the genus of a pair of maps of inverse ANR-systems associated with f and g. If f,g are maps of CW-complexes, then the \({\mathfrak F}\)-genus equals the shape \({\mathfrak F}\)- genus.
The author states that in a forthcoming paper he will obtain necessary and sufficient conditions, involving the shape genus of certain maps, for an n-dimensional compactum to be embeddable up to shape into \(E^{2n}\).
Reviewer: L.Rubin
55P55 Shape theory
54C56 Shape theory in general topology
57N25 Shapes (aspects of topological manifolds)
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