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.
