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Multihomotopy, Čech Spaces of loops and Shape Groups Rodríguez Sanjurjo, José Manuel 515.143 multi-net upper-semicontinuous multi-valued maps shape morphisms shape groups spaces of loops Cech space of loops Topología 1210 Topología 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. Depto. de Álgebra, Geometría y Topología Fac. de Ciencias Matemáticas TRUE pub 2023-06-20T18:46:25Z 2023-06-20T18:46:25Z 1994-09 journal article https://hdl.handle.net/20.500.14352/58575 0024-6115 10.1112/plms/s3-69.2.330 metadata only access Oxford University Press (OUP)