A celebrated theorem of Weiss and Williams expresses (in a range of homotopical degrees) the difference between the spaces of diffeomorphisms and block diffeomorphisms of a manifold in terms of its algebraic K-theory. In this talk, I will present an analogous result for embedding spaces. Namely, for M a manifold of dimension at least 5 and P submanifold of M of codimension at least 3, we describe the difference between the spaces of block and ordinary embeddings of P into M as a certain infinite loop space involving the relative algebraic K-theory of the
pair (M, M − P ). The range of degrees in which this description applies is the so-called concordance embedding stable range which, by recent developments of Goodwillie–Krannich–Kupers, is far beyond that of the aforementioned theorem of Weiss–Williams.
I will also explain how one can use this result to give a full description of the homotopy type (away from 2 and roughly up to the concordance embedding stable range) of the space of long knots of codimension at least 3, that is, embeddings rel boundary of the p-disk into the d-disk for d − p > 2 and d > 4.