Piecewise linear approximations of embeddings of cells and spheres in codimensions higher than two

Recently the paper of Homma (RZhMat., 1968, 5A492) 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 (RZhMat., 1964, 10A298).
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