Bordered Floer homology and incompressible surfaces
Annales de l'Institut Fourier, Volume 69 (2019) no. 4, p. 1525-1573

We show that bordered Heegaard Floer homology detects homologically essential compressing disks, and that bordered-sutured Floer homology detects partly boundary-parallel tangles and bridges, in natural ways. For example, there is a bimodule Λ so that the tensor product of CFD ^(Y) and Λ is Hom-orthogonal to CFD ^(Y) if and only if the boundary of Y admits a homologically essential compressing disk. In the process, we sharpen a nonvanishing result of Ni’s. We also extend Lipshitz–Ozsváth–Thurston’s “factoring” algorithm for computing HF ^  to compute bordered-sutured Floer homology, making both results on detecting essential incompressibility practical. In particular, this makes computing Zarev’s tangle invariant manifestly combinatorial.

Nous montrons que l’homologie de Heegaard Floer bordée détecte les disques de compression homologiquement essentiels et que l’homologie de Floer bordée-suturée détecte les enchevêtrements partiellement parallèles au bord, de manière naturelle. Par exemple, il y a un bimodule Λ tel que le produit tensoriel de CFD ^(Y) et Λ est Hom-orthogonal à CFD ^(Y) si et seulement si le bord de Y admet un disque du compression homologiquement essentiel. Nous affinons aussi un résultat de Ni sur la non annulation de l’homologie de Heegaard Floer et nous étendons l’algorithme “factorisation” de Lipshitz–Ozsváth–Thurston  pour calculer l’homologie de Floer bordée-suturée, de sorte que les deux résultats sur la détection des surfaces incompressibles sont effectifs. En particulier, nous montrons que le calcul de l’invariant de l’enchevêtrement de Zarev est combinatoire.

Received : 2017-08-31
Revised : 2018-04-16
Accepted : 2018-06-12
Published online : 2019-09-16
DOI : https://doi.org/10.5802/aif.3276
Classification:  57M27,  53D40
Keywords: Heegaard Floer homology, bordered Floer homology, sutured manifolds, incompressible surfaces
@article{AIF_2019__69_4_1525_0,
     author = {Alishahi, Akram and Lipshitz, Robert},
     title = {Bordered Floer homology and incompressible surfaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {4},
     year = {2019},
     pages = {1525-1573},
     doi = {10.5802/aif.3276},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_4_1525_0}
}
Bordered Floer homology and incompressible surfaces. Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1525-1573. doi : 10.5802/aif.3276. https://aif.centre-mersenne.org/item/AIF_2019__69_4_1525_0/

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