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Bimodules in bordered
Heegaard Floer homology
Robert Lipshitz, Peter S Ozsváth and Dylan P Thurston |
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Geometry & Topology 19 (2015) 525–724
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Correction: 10.2140/gt.2024.28.1001
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Abstract |
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Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface an algebra and to a three-manifold with boundary identified with a module over . In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between and the boundary of tensors the bordered invariant with a suitable bimodule over . These bimodules give an action of a suitably based mapping class group on the category of modules over . The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated open book decomposition. In the course of establishing these results, we also calculate the homology of . We also prove a duality theorem relating the two versions of the –manifold invariant. Finally, in the case of a genus-one surface, we calculate the mapping class group action explicitly. This completes the description of bordered Heegaard Floer homology for knot complements in terms of the knot Floer homology. |
Keywords
Floer homology, $3$–manifolds, Heegaard diagrams, mapping
class group, Hochschild homology
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Mathematical Subject Classification 2010
Primary: 57R57
Secondary: 53D40
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References |
Publication
Received: 1 July 2011
Revised: 23 April 2014
Accepted: 5 June 2014
Published: 10 April 2015
Proposed: Yasha Eliashberg
Seconded: Ciprian Manolescu, Tomasz Mrowka
Correction: 13 March 2024
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Authors |
| Robert Lipshitz | |
| Department of Mathematics Columbia University MC 4425 2990 Broadway New York, NY 10027 USA |
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| Peter S Ozsváth | |
| Department of Mathematics Princeton University Fine Hall, Washington Road Princeton, NY 08540 USA |
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| Dylan P Thurston | |
| Department of Mathematics Indiana University 831 E Third St Bloomington, NY 47405 USA |
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