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Bordered Floer homology for manifolds with torus boundary via immersed curves.
- Source :
-
Journal of the American Mathematical Society . Apr2024, Vol. 37 Issue 2, p391-498. 108p. - Publication Year :
- 2024
-
Abstract
- This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M is such a manifold, we show that the type D structure \widehat {CFD}(M) may be viewed as a set of immersed curves decorated with local systems in \partial M. These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism h between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of \widehat {HF} decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of \widehat {HF}. In particular, it follows that a prime rational homology sphere Y with \widehat {HF}(Y)<5 must be geometric. Other results include a new proof of Eftekhary's theorem that L-space homology spheres are atoroidal; a complete characterization of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots. [ABSTRACT FROM AUTHOR]
- Subjects :
- *FLOER homology
*TORUS
*KNOT theory
*ISOMORPHISM (Mathematics)
*GLUE
Subjects
Details
- Language :
- English
- ISSN :
- 08940347
- Volume :
- 37
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Journal of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 175027057
- Full Text :
- https://doi.org/10.1090/jams/1029