Approximating cellular maps by homeomorphisms

Abstract

M -+ N is a limit of homeomorphisms if and only if each point preimage f-‘(y), y E N, is compact and contractible. Contractibility is here meant in the weak sense of Borsuk [lo], and is equivalent to the terms shape of a point or cell-like or UV, . The space H(M, N) of homeomorphisms M to N of homeomorphic manifolds that are compact without boundary (i.e. closed), and of dimension m > 0, is never closed in the space of continuous maps M to N for the compact-open topology. In fact M. Brown has proved [12] that if X is any compacturn in M which is cellular in M (i.e. X has small open neighbourhoods that are open m-cells), then M/X (= M with X collapsed to a point) is a manifold, call it N, and the quotient map M -+ N is a limit of homeomorphisms. An example of a compacturn cellular in the plane R2 is the closure of {(x, sin(l ix)) [ 0 < x I I}. On the other hand R. Finney [22] has observed that any mapf: M + N, which is a limit of homeo- morphisms, is cellular in the sense that f-‘(y) is cellular for all y in N. The proof is a pleasant exercise. One is thus led to conjecture that the limits of homeomorphisms are pre- cisely the cellular maps (not less). And the conjecture is made more significant by the fact from engulfing [35] [42] that, if m # 3,4,