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A
Weiss–Williams theorem for spaces of embeddings and the
homotopy type of spaces of long knots
Samuel Muñoz-Echániz |
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Geometry & Topology 30 (2026) 701–780
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Abstract |
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We establish a pseudoisotopy result for embedding spaces in the line of that of Weiss and Williams for diffeomorphism groups. In other words, for an embedding of codimension at least , we describe the difference in a range of homotopical degrees between the spaces of block and ordinary embeddings of into as a certain infinite loop space involving the relative algebraic -theory of the pair . This range of degrees is the so-called concordance embedding stable range, which, by recent developments of Goodwillie, Krannich, and Kupers, is far beyond that of the aforementioned theorem of Weiss and 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 . 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 -theory that may be of independent interest. |
Keywords
embedding space, pseudoisotopy, long knot
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Mathematical Subject Classification
Primary: 18F50, 19D10, 57R40, 57R80, 58D10
Secondary: 57S05
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References |
Publication
Received: 17 May 2024
Revised: 20 August 2025
Accepted: 26 September 2025
Published: 16 March 2026
Proposed: Nathalie Wahl
Seconded: Haynes R Miller, Roman Sauer
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Authors |
| Samuel Muñoz-Echániz | |
| Centre for Mathematical
Sciences University of Cambridge Cambridge United Kingdom |
Mathematics Department Massachusetts Institute of Technology Cambridge, MA United States |
| © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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