Vol. 46, No. 1, 1973

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On extending isotopies

William Cutler

Vol. 46 (1973), No. 1, 31–35

Let K be a locally compact metric space. An isotopy on K is a continuous family of homeomorphisms ht : K K for t I such that h0 = id. Let (K) denote the space of isotopies of K with C 0 topology. Conjecture: Let X be metric and Y a closed subset of X. Then every map f : Y →ℐ(K) can be continuously extended to X. The conjecture is proved for the following cases: (1) K is a l-complex, (2) K is compact and X is finite-dimensional, (3) K is compact, Y is compact and finite-dimensional, and X is separable, and (4) Y is of type 1 in a compact space X. Y is of type 1 in X if the closure of the set of points of X which do not have a unique closest point in Y does not intersect Y .

Mathematical Subject Classification 2000
Primary: 54C20
Received: 7 February 1972
Revised: 24 May 1972
Published: 1 May 1973
William Cutler