Shape index in metric spaces

Volume 176 / 2003

Francisco R. Ruiz del Portal, José M. Salazar Fundamenta Mathematicae 176 (2003), 47-62 MSC: 37B30, 54H20, 54C56. DOI: 10.4064/fm176-1-4


We extend the shape index, introduced by Robbin and Salamon and Mrozek, to locally defined maps in metric spaces. We show that this index is additive. Thus our construction answers in the affirmative two questions posed by Mrozek in [12]. We also prove that the shape index cannot be arbitrarily complicated: the shapes of $q$-adic solenoids appear as shape indices in natural modifications of Smale's horseshoes but there is not any compact isolated invariant set for any locally defined map in a locally compact metric ANR whose shape index is the shape of a generalized solenoid. We also show that, for maps defined in locally compact metric ANRs, the shape index can always be computed in the Hilbert cube. Consequently, the shape index is the shape of the inverse limit of a sequence $\{P_n, g_n\}$ where $P_n=P$ is a fixed ANR and $g_n=g: P \rightarrow P$ is a fixed bonding map.


  • Francisco R. Ruiz del PortalDepartamento de Geometría y Topología
    Facultad de Matemáticas
    Universidad Complutense de Madrid
    Madrid 28040, Spain
  • José M. SalazarDepartamento de Matemáticas
    Universidad de Alcalá
    Alcalá de Henares
    Madrid 28871, Spain

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