Проблема вложения многообразий в Евклидовы пространства

A clear understanding of topology of higher-dimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higher-dimensional topology in a way which makes clear the visual and intuitive part of the constructions and the arguments.
In particular, we show how abstract algebraic constructions appear naturally in the study of geometric problems. Before giving a general construction, we illustrate the main ideas in simple but important particular cases, in which the essence is not veiled by technicalities. More specifically, we present several classical and modern results on the embeddability of manifolds in Euclidean space.
We state many concrete results (in particular, recent results in lower dimensions). Their statements (but not proofs!) are simple and accessible to non-specialists. We state some results for the case of $\mathbb{R}P^n$ and $\mathbb{C}P^n$. We outline a general approach to embeddings via the classical van Kampen-Shapiro-Wu-Haefliger-Weber 'deleted product' obstruction. This approach reduces the question of embeddability to the existence of equivariant maps, and so implies the above concrete results. We describe the revival of interest in this beautiful branch of topology, by presenting new results in this area: a generalization the Haefliger-Weber embedding theorem below the metastable dimension range and examples showing that other analogues of this theorem are false outside the metastable dimension range. The talk is based on the paper https://arxiv.org/abs/math/0604045.