A Weiss-Williams theorem for spaces of embeddings and the homotopy type of spaces of long knots

Abstract

We establish a pseudoisotopy result for embedding spaces in the line of that of Weiss and Williams for diffeomorphism groups. In other words, for $P\subset M$ a codimension at least three embedding, we describe the difference in a range of homotopical degrees 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 $(M,M-P)$. This range of degrees 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. We use this result to obtain split fibre sequences in the concordance embedding stable range, with explicit, analysable base and fibre, which determine the homotopy type of spaces of long knots of codimension at least 3. This leads to explicit computations of homotopy groups, including torsion information, in that range. In doing so, we carry out an extensive analysis of certain geometric involutions in algebraic $K$-theory that may be of independent interest.