On the algebra of cornered Floer homology (Q2879004)

This paper presents two preliminary constructions which point toward a cornered theory for Heegaard--Floer (HF) homology. Precisely, it sets up a bordered theory for bordered HF homology, and a cornered theory for the simpler (but non topologically interpreted) planar grid HF homology.NEWLINENEWLINEIn [Ann. Math. 159 (2), 1027--1158 (2004; Zbl 1073.57009)], \textit{P. Ozsváth} and \textit{Z. Szabó} defined, for any pointed Heegaard diagram \(\Sigma\) of a closed 3--manifold \(Y\), a chain complex whose homology is an invariant of \(Y\). Its differential counts disks in a symmetric product of \(\Sigma\) satisfying some global and analytic pseudo-holomorphic conditions. Based on a reformulation due to R. Lipshitz which substitute disks with surfaces in a thickening of \(\Sigma\), \textit{R. Lipshitz, P. Ozsváth} and \textit{D. Thurston} gave in [``Bordered Heegaard Floer homology: Invariance and pairing'', (2008) \url{arXiv:0810.0687}] a bordered HF theory (over \(\mathbb{F}_2\)) for 3--manifolds with boundary. It can be understood as cutting every object (3--manifolds, Heegaard diagrams, pseudo-holomorphic surfaces) in two halves. The theory starts with a closed surface \(F\), described by a pointed Heegaard diagram \(\mathcal{Z}\) over a circle, and a 3--manifold \(Y\) with boundary \(F\), described by a pointed Heegaard diagram \(\Sigma\) bordered by \(\mathcal{Z}\); it definesNEWLINENEWLINE- a graded differential algebra \(\mathcal{A}(\mathcal{Z})\) which is not an invariant of \(F\) but depends on the choice of \(\mathcal{Z}\);NEWLINENEWLINE- a graded differential algebra \(\text{CFD}(\Sigma)\) over \(\mathcal{A}(\mathcal{Z})\);NEWLINENEWLINE- a graded \(\mathcal{A}_\infty\)--algebra \(\text{CFA}(\Sigma)\) over \(\mathcal{A}(\mathcal{Z})\).NEWLINENEWLINEThe differential in \(\text{CFD}(\Sigma)\) counts pseudo-holomorphic surfaces over \(\Sigma\) and records in \(\mathcal{A}(\mathcal{Z})\) how the surfaces meet \(\mathcal{Z}\), whereas the \(\mathcal{A}_\infty\)--multiplication in \(\text{CFA}(\Sigma)\) counts pseudo-holomorphic surfaces over \(\Sigma\) which meet \(\mathcal{Z}\) in a certain way specified by elements of \(\mathcal{A}(\mathcal{Z})\). The first point can be seen as some HF theory for closed surface, applied to the interface of a cut. There is a pairing theorem which enables a backward gluing: it states that the HF homology of \(Y=Y_1\cup_F Y_2\), where \(F\) is a closed surface, can be recovered as the homology of a certain tensor product \(\text{CFD}(\Sigma_1)\widetilde{\otimes}\text{CFA}(\Sigma_2)\) over \(\mathcal{A}(\mathcal{Z})\), where \(\mathcal{Z}\), \(\Sigma_1\) and \(\Sigma_2\) are respectively pointed Heegaard diagrams for \(F\), \(Y_1\) and \(Y_2\). Lipshitz--Ozsváth--Thurston theory naturally handles multiple parallel (\textit{i.e.} disjoint) cuts.NEWLINENEWLINEThe present paper investigates orthogonal cuts: the intersection of the cuts produces then a codimension 2 locus called corner. Such a corner splits in halves both surfaces at the interface of the cuts. Any cornered HF theory for closed 3--manifolds requires hence a bordered HF theory for closed surfaces. This is the first achievement of the paper. The authors start with a surface with circular boundary, described by a pointed Heegaard diagram \(\mathcal{Z}\) over an interval, and defineNEWLINENEWLINE- a differential graded 2-algebra \(\mathfrak{N}\), namely the nil-Coxeter sequential 2-algebra, associated to the circle;NEWLINENEWLINE- a differential graded algebra-module \(\mathcal{T}(\mathcal{Z})\) over \(\mathfrak{N}\);NEWLINENEWLINE- a graded algebra-module \(\mathcal{B}(\mathcal{Z})\) over \(\mathfrak{N}\).NEWLINENEWLINEThen they prove a pairing theorem which recovers \(\mathcal{A}(\mathcal{Z})\) as some tensor product \(\mathcal{T}(\mathcal{Z}_1)\odot\mathcal{B}(\mathcal{Z}_2)\) over \(\mathfrak{N}\), where \(\mathcal{Z}_1\cup \mathcal{Z}_2\) is a decomposition of \(\mathcal{Z}\).NEWLINENEWLINENow, a complete cornered theory would contain HF theories for each of the four cut pieces and pairing theorems for gluing them pairwise in order to recover the \(\text{CFD}\) and \(\text{CFA}\) structures. Such a theory in the simplified context of HF homology for planar grids is the second achievement of the paper. Planar grids encode links in \(S^3\) and the associated homology is a combinatorial object, closely related to the combinatorial link Floer homology developed in [\textit{C. Manolescu} et al., Geom. Topol. 11, 2339--2412 (2007; Zbl 1155.57030)], but which does not carry topological invariance. It has been developed in [\textit{R. Lipshitz} et al., in: Proceedings of the 15th Gökova geometry-topology conference, Gökova, Turkey, May 26--31, 2008. Cambridge, MA: International Press. 91--119 (2009; Zbl 1201.57025)] as a toy model for bordered HF homology, where all higher multiplications in the \(\mathcal{A}_\infty\)--structure vanish. So-called \textit{nice} Heegaard diagrams for 3--manifolds is another combinatorial situation where higher multiplications vanish and the approach adopted in the paper can be applied in this situation as well; however, proving topological invariance for the structures would be then a delicate process.NEWLINENEWLINEThe paper is organized as follows. Section 2 introduces all the algebraic structures mentioned above. Section 3 deals with the strands algebra \(\mathcal{A}(N)\), a graded differential algebra defined for each non negative integer \(N\), which will contain \(\mathcal{A}(\mathcal{Z})\) as a subalgebra. After having defined each summand, it proves a decomposition theorem of the form \(\mathcal{A}(N+N')\cong\mathcal{T}(N)\odot\mathcal{B}(N')\) over \(\mathfrak{\mathcal{Z}}\), which prefigures the pairing theorem for \(\mathcal{A}(\mathcal{Z})\). Section 4 presents all the relevant Heegaard diagram notions while section 5 defines \(\mathcal{T}(\mathcal{Z})\), \(\mathcal{B}(\mathcal{Z})\) as sub-objects of \(\mathcal{T}(N)\), \(\mathcal{B}(N)\) and proves the desired pairing theorem. Section 6 recalls all the material for planar grids homology from [loc. cit.]. Section 7 and 8 are the most technical sections; they cut a planar grid in four quadrants and define explicitely the relevant algebra for each of them. Then, it is proved combinatorially that all the needed relations are satisfied. Gluing theorems arise then as natural consequences of the definitions.