Embedding compacta in Euclidean space of dimension $5$, $4$ and $3$

In this paper results concerning the notion of dimension of an embedding, earlier obtained for embeddings of compacta in Euclidean space $E_n$ for $n\geqslant6$, are extended to the case $n=5$. For $n=4$ a reduction of the main problem to the so-called open four-dimensional Poincaré conjecture is given, and some sufficient conditions are given for $n=3$.
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