Bordered Floer homology for manifolds with torus boundary via immersed curves

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 CFD(M ) may be viewed as a set of immersed curves decorated with local systems in ∂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 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 HF . In particular, it follows that a prime rational homology sphere Y with 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 characterisation 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.

Heegaard Floer homology and cosmetic surgeries in $S^3$

“…We will primarily be interested in a weaker form of the invariant, which we call Γ(K) and which is equivalent to the U V = 0 truncation of the knot Floer complex. The U V = 0 truncation of knot Floer homology is also equivalent to bordered Floer homology of the knot complement, and an immersed curve description of this invariant is due to the author, Rasmussen, and Watson [8,9] (the case of knot complements is discussed specifically in [9, Section 4]). In particular, the invariant denoted Γ(K) in this paper agrees with HF (M ) with M = S 3 \ ν(K) in the notation of [8,9].…”

“…The U V = 0 truncation of knot Floer homology is also equivalent to bordered Floer homology of the knot complement, and an immersed curve description of this invariant is due to the author, Rasmussen, and Watson [8,9] (the case of knot complements is discussed specifically in [9, Section 4]). In particular, the invariant denoted Γ(K) in this paper agrees with HF (M ) with M = S 3 \ ν(K) in the notation of [8,9]. For readers unfamiliar with bordered Floer homology, a bordered free construction of the immersed curves Γ(K) will appear in a forthcoming paper by the author [6].…”

“…The selected intersection points should be interpreted as places where a traveler along the negatively sloped segment the curve in the boxed region depicted in Figure 1(a), is allowed to make a left turn. In the language of [8], this means that we extend the curve γ i to an immersed train track and add two (oriented) edges near the selected intersection point, as shown in Figure 1(b). Decorating a chosen subsection of intersection points by adding train track edges in this way determines a k i × k i invertible matrix with coefficients in F which counts immersed paths from the left to the right of the boxed region.…”

“…In particular, we consider the isomorphism type of HF as an absolutely graded vector space, which amounts to keeping track of the grading for each generator in addition to the rank. This is facilitated by a recent reinterpretation of Heegaard Floer invariants for manifolds with torus boundary in terms of collections of immersed curves due to the author, Rasmussen, and Watson [8,9]. In particular, this provides a combinatorial framework which makes comparing gradings for surgeries on knots easier.…”

Instanton L-spaces and splicing

“…A Heegaard Floer analogue of Theorem 1.1 was first established by Hedden and Levine in [HL16], and later generalized by Eftekhary [Eft15,Eft20] and then Hanselman, Rasmussen, and Watson [HRW17]. Their approaches used bordered Heegaard Floer homology or something similar.…”