The closure of the space of homeomorphisms on a manifold

Abstract

The space, H(M), of all mappings of the compact manifold M onto itself which can be approximated arbitrarily closely by homeomorphisms is studied. It is shown that H(M) is homogeneous and weakly locally contractible. If M is a compact 2-manifold without boundary, then H(M) is shown to be locally contractible. 1. Let M be a compact manifold and H(M) denote the space of all homeomorphisms of M onto itself. We shall study the space, H(M), of all continuous functions of M onto itself which can be approximated arbitrarily closely by elements of H(M). All function spaces on compact spaces will be assumed to have the supremum metric,p; i.e., if X and Y are spaces with d the metric on Y and / and g are functions from X into Y, then p(f, g) supx,xtd(f(x), g(x))I. Since M is compact, the topology thus generated agrees with the compact-open topology. A mapping of an n-manifold, Mn, onto itself is said to be cellular if for each y e Mn, f 1(y) can be expressed as the intersection of a nested sequence of ncells. Armentrout (n 5) [20] have recently shown that H(Mn), n g 4, is precisely the space of all cellular mappings of Mn onto itself. Hence most of the results of this paper could be stated in terms of spaces of cellular mappings. Cellular mappings have been studied extensively (cf., Lacher [15], [16]). Let H8(M) denote the space of all homeomorphisms of M onto itself which equal the identity when restricted to the boundary of M and, following our previous notation, let H11(M) denote the space of all continuous functions of M onto itself which can be approximated arbitrarily closely by elements of H8(M). We shall state some of the major results concerning H(M) and H8(M) and then indicate which of the analogous theorems can be proven for H(M) and H18(M): Received by the editors December 20, 1972 and, in revised form, July 9, 1973. AMS (MOS) subject classifications (1970). Primary 54H15, 57E05; Secondary 57A60, 57A20.