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In this paper the group of homeomorphisms of an arbitrary topological manifold is considered, with either the compact-open, uniform (relative to a fixed metric), or majorant topology. In the latter topology, a basis of neighborhoods of the identity is given by the strictly positive functions on the manifold, a homeomorphism being in the neighborhood determined by such a function if it moves each point less than the value of this function at the point. The main result of the paper is the proof of the local contractibility of the group of homeomorphisms in the majorant topology. Examples are easily constructed to show that this assertion is false for the other two topologies for open manifolds. In the case of a compact manifold the three topologies coincide. In conclusion a number of corollaries are given; for example, if a homeomorphism of a manifold can be approximated by stable homeomorphisms then it is itself stable.

Recently the paper of Homma (Yokohama Math. J. 14 (1966), 47-54; MR 36 #892) which implies the possibility of piecewise linear approximation of piecewise linear manifolds in codimensions higher than two was found to contain an error, so that it is at present unclear whether the proof of this result can be completed using Homma's method. The present paper gives a proof of this result for the case of the elementary manifolds (cells and spheres), thus preserving the validity of two recently proved results whose proof were based on Homma's theorem. The method of proof used in this paper differs from Homma's method and is close to Connell's proof for approximation of stable homeomorphisms (Ann. of Math. (2) 78 (1963), 326-338; MR 27 #4238). Bibliography: 19 items

It is shown that an arbitrary locally flat imbedding of one piecewise linear manifold into another of codimension greater than two can be changed to a piecewise linear imbedding by means of an arbitrarily small isotopy of the ambient manifold. Bibligraphy: 5 entries.