On isotopic and discrete realizations of maps of an $n$-dimensional sphere in Euclidean space

In this paper we consider questions of whether a compact space can be embedded in a Euclidean space. The problem of embedding an '$S^n$-like' compact space in $\mathbb R^{2n}$ is solved affirmatively under certain restrictions on the dimension $n$. We clarify the relations between the realization problem and areas of homotopy theory and differential topology.