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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
##### MSC:
 55P55 Shape theory 54C56 Shape theory in general topology 57N25 Shapes (aspects of topological manifolds)
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