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In this paper, we prove a number of theorems which give some insight into the structure of differentiable manifolds. The methods, results and some notation of 13, hereafter referred to as GPC, and 12 will be used. These two papers and 14 can be considered as a starting point for this one. The main theorems in these papers are special cases of the theorems here. Among the most important theorems in this paper are 1. 1 and 6. 1. Some conversations with A. Haefliger were helpful in the preparation of parts of this paper. Everything will be considered from the differentable, equivalently C*, point of view; manifolds, imbeddings, and isotopes will be C*.
Stephen T. Smale (Sun,) studied this question.