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Summary of A. Skop enkov's prop osal This proposal concerns the classical Knotting Problem in topology: classify embeddings of a given space into another given space up to isotopy. This problems have played an outstanding role in the development of topology. Various methods for the investigation of the Knotting Problems were created by classical figures. The Knotting Problem is known to be hard. E.g. for the best known specific case of codimension 2 embeddings a complete classification is neither known nor expected. The Knotting Problem is most interesting for manifolds of dimension at most 4, because embeddings of such manifolds often appear in other branches of mathematics and its applications. I work in the smooth category unless PL (piecewise linear) category us explicitly mentioned. Classical results of Wu, Haefliger, Hirsch (1960-s) on embeddings of n-dimensional manifolds into Rm have the metastable dimension restriction 2m > 3n + 3. In particular, in low dimensions Haefliger and Hirsch classified embeddings of 3-dimensional manifolds into Rm for m 7, and of 4-dimensional manifolds into Rm for m 8. The main intention of this research proposal is to classify embeddings for 2m 3n + 3 and for closed connected manifolds. For N not a homology sphere until 2005 no classification was known, in spite of the existence of interesting partial results, results in the PL category and approaches of Browder-Wall and Goodwillie-Weiss. Embeddings S n S m for m > n + 2 were classified by A. Haefliger in 1960s. There is the `connected sum' action # of the group of embeddings S n S m on the set of embeddings N S m for a closed connected orientable n-manifold N . The quotient set of this action was known for some cases including the case m = 6 = 2n and the case N simply-connected, m = 7 = 2n - 1; there remained to find the orbits of #. For N not a homology sphere until 2005 no description of the orbits was known. A description of orbits for these cases appeared in papers by Crowley and myself in 20082010. They yielded a classification of embeddings for 3-dimensional manifolds in R6 and for simply-connected 4-dimensional manifolds in R7 . D. Crow ley and I plan to obtain a classification of embeddings of non-simply-connected 4manifolds in R7 . This is a significant new step as it is explained in the proposal. We also plan to obtain a piecewise linear analogue of this result. My contribution to our joint plan would be a description of the quotient set of # and the `lower estimation' in the description of the orbits of #. I also plan to study the fol lowing Compression Problem for n = 4 and m = 6, 7: characterize embeddings of n-manifolds into Rm+1 that are isotopic to embeddings into Rm . Many interesting examples of embeddings are embeddings S p в S q S m . A classification of such embeddings is a natural next step (after the link theory and the classification of embeddings of highly-connected manifolds) towards classification of embeddings of arbitrary manifolds. Assume that 1 p q . A classification for m 2q + p + 1 and 2m 3q + 3p + 4 was obtained by Haefliger-Hirsch in 1963, and for m q + 2p + 3 and 2m 3q + 2p + 3 by myself recently. I plan to obtain a classification of embeddings S p в S q S m for 1 p q and m q + 2p + 3. The classification would be in terms of homotopy groups of spheres and embeddings Dp+1 в S q S m and S p+q S m which are easier to describe. I plan to define an action of the homology group of a manifold N on the set of embeddings N S m . This would require a partial solution of the following problem of E. Rees: describe the action of self-diffeomorphisms of S p в S q on isotopy classes of embeddings S p в S q S m . 1