Vol. 108, No. 1, 1983

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Fixed point sets of homotopies

Helga Schirmer

Vol. 108 (1983), No. 1, 191–202
Abstract

In recent years it has been shown that many spaces have the so-called complete invariance property, i.e. that every closed and non-empty subset of them can be realized as the fixed point set of a continuous selfmap. Here a related result is obtained for homotopies H : X ×I X rather than selfmaps of a space X. The theorem proved here states that if P is a compact and connected polyhedron without local cut points and K P × I a closed set which contains a continuum intersecting both X × 0 and X × 1, then there exists a homotopy H : P × I P with fixed point set K.

Mathematical Subject Classification 2000
Primary: 55M20
Secondary: 54H25
Milestones
Received: 30 October 1981
Revised: 31 March 1982
Published: 1 September 1983
Authors
Helga Schirmer