Multihomotopy, Čech Spaces of loops and Shape Groups

Recently, the author has given an alternate (and intrinsic) description of the shape category of metric compacta, based on the notion of multi-nets F:X→Y. These are defined as sequences (Fk) of upper semicontinuous multivalued mappings Fk:X→Y, whose values Fk(x), x∈X, have diameters tending to 0. Shape morphisms X→Y are defined as homotopy classes of multi-nets [J. M. R. Sanjurjo, Trans. Amer. Math. Soc. 329 (1992), no. 2, 625–636. In the present paper the author considers the set N(X,Y) of all multi-nets and endows it with a T0-topology. It is proved that two multi-nets are homotopic if and only if they belong to the same path-component of N(X,Y). A certain subspace of N(I,X), I=[0,1], is the Čech space of loops Ωˇ(X,x0). Its path components can be identified with the first shape group πˇ1(X,x0). The author also shows that the nth shape group πˇn(X,x0) coincides with a certain subgroup of the fundamental group of the iterated loop space Ωˇn−1(X,x0). These results assume a simple form if they are applied to internally movable compacta [J. Dydak, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 1, 107–110 and internal FANRs [V. F. Laguna and J. M. R. Sanjurjo, Topology Appl. 17 (1984), no. 2, 189–197. Finally, the author considers continuous flows π:M×R→M, where M is a locally compact ANR. It is proved that every asymptotically stable compact set X⊆M is shape dominated by a compact polyhedron, i.e., X is an FANR. In a remark the author points out that this theorem has also been obtained independently by B. Günther and J. Segal [Proc. Amer. Math. Soc. 119 (1993), no. 1, 321–329