Distance-Two Surgery: Exact Triangle in Floer Homology

• Distance-Two Surgery Exact Triangle is a distinguished long exact sequence in Floer homology that links invariants of 3-manifolds obtained by Dehn surgeries differing by distance two.
• It employs non-orientable band maps, holomorphic triangle counts, and 2-handle cobordism maps to reveal intertwined algebraic and geometric features in knot and 3-manifold theories.
• The triangle is pivotal for computing correction terms and distinguishing homology 3-spheres, particularly in surgery settings on alternating knots and lens space surgeries.

The distance-two surgery exact triangle is a distinguished long exact sequence in Floer homology theories associated with 3-manifolds and knots, relating the invariants of manifolds obtained by three distinct Dehn surgeries whose associated slopes differ by distance two in the homological coordinates of the knot complement. In Heegaard Floer, instanton Floer, and Pin(2)-monopole Floer settings, the exact triangle captures subtle interdependencies between homology groups under 2-handle attachment and Band maps, including the algebraic and geometric invariants arising from non-orientable and merge band moves. The triangle construction enables computation of correction terms and identification of homological features that distinguish families of 3-manifolds, especially after surgery on alternating knots (Nahm, 2 Jan 2025, Lin, 2015).

1. Formal Statement of the Distance-Two Surgery Exact Triangle

Let $Y$ be a closed, oriented 3-manifold, $K \subset Y$ a knot with meridian $\mu$ and arbitrary framing $\lambda$. Define the associated surgeries:

• $Y_\infty(K) = Y$ (the manifold before surgery),
• $Y_\lambda(K)$ obtained by $\lambda$-surgery,
• $Y_{\lambda+2\mu}(K)$ by $(\lambda+2\mu)$-surgery.

In Heegaard Floer theory over $\mathbb{F}_2[[U]]$, there is an exact triangle of modules:

$$\cdots \longrightarrow HFL'^{-}(Y,K) \xrightarrow{F_1} HF^{-}(Y_\lambda(K)) \xrightarrow{F_2} HF^{-}(Y_{\lambda+2\mu}(K)) \xrightarrow{F_3} HFL'^{-}(Y,K) \longrightarrow \cdots$$

Here,

• $HFL'^{-}(Y,K)$ is the unoriented knot Floer homology,
• $HF^{-}(-)$ is the standard Heegaard Floer homology for 3-manifolds,
• $F_1$, $F_3$ are non-orientable (“band”) maps,
• $F_2$ is the 2-handle cobordism map (Nahm, 2 Jan 2025).

In Pin(2)-monopole Floer homology, consider three manifolds $Y_0$, $Y_1$, $Y_2$ obtained by Dehn fillings along slopes $\gamma_i$, $\gamma_{i+1}$, $\gamma_{i+2}$ such that the intersection numbers are $-1$ mod 3 and collectively have distance two. There is an exact triangle:

$$\cdots \xrightarrow{f_0} \widetilde{HS}_*(Y_1) \xrightarrow{f_1} \widetilde{HS}_*(Y_2) \xrightarrow{f_2} \widetilde{HS}_*(Y_0)[-1] \xrightarrow{f_0} \cdots$$

Maps $f_i$ arise from counting solutions on spin-cobordisms, while the third (from non-spin handle) uses pentagon stretching and null-homotopy arguments (Lin, 2015).

2. Chain Complexes and Module Structure

The construction involves specialized Floer chain complexes:

• $CFL'^{-}(Y,K)$: Built from a doubly-pointed Heegaard diagram $(\Sigma, \alpha, \beta, \{w, z\})$. The chain differential counts holomorphic disks with multiplicity one, weighting both basepoints $w$ and $z$ by $U^{1/2}$, forming an $\mathbb{F}_2[[U]]$-module. Homology yields $HFL'^{-}(Y,K)$.
• $CF^{-}(Y_\lambda(K))$ and $CF^{-}(Y_{\lambda+2\mu}(K))$: Constructed from single-pointed Heegaard diagrams for each surgered manifold.

In Pin(2)-monopole Floer homology, chain complexes are $R$-modules with $R = \mathbb{F}_2[[V]][Q]/(Q^3)$, using basepoint insertions to encode the completed module structure. Three primary flavors $$(\widetilde{HS}_*, \widehat{HS}_*, \overline{HS}_*)$$ assemble into the “to-from-bar” triangle (Lin, 2015).

3. Construction and Properties of Maps

The triangle is defined via three types of maps:

• Non-orientable band map ($F_1$): Realized by a triple-Heegaard diagram $(\Sigma, \alpha, \beta, \gamma, \{w, z\})$, encoding a non-orientable band from $K$ to the core circle of $\lambda$-surgery. On chains, it is a holomorphic triangle map $\mu_2(-, \Theta_B)$, with $\Theta_B$ the top-degree cycle from the band (Nahm, 2 Jan 2025).
• 2-handle cobordism map ($F_2$): Defined by counting holomorphic triangles in the diagram representing $\lambda \rightarrow \lambda+2\mu$ handle attachment.
• Merge-band map ($F_3$): Via a triple diagram for the dual band from the $\lambda+2\mu$-surgered manifold to the knot. The algebraic definition involves projection to submodules matching the merge algebra.

In Pin(2)-monopole theory, for surgery triple $(Y_0,Y_1,Y_2)$, maps are associated with spin-cobordisms when handles are spin. The non-spin handle produces a null-homotopic map, filled in via mapping cone and spectral sequence considerations (pentagon stretching), uniquely determined by vanishing of the blow-up map and functoriality requirements (Lin, 2015). The triangle property is ensured via the mapping cone lemma: compositions of any two maps are null-homotopic.

4. Gradings, Spin$^c$ Splittings, and Algebraic Features

All complexes in the exact triangle carry a Maslov grading; $\delta$-grading is relative $\mathbb{Z}$ for $CFL'^{-}(Y,K)$, and the $U$-action shifts it by $2$. There is a Spin$^c$ splitting at each vertex, refining the triangle: summands split over $\operatorname{Spin}^c(Y\setminus K)/2\mu$, and all maps respect this structure (Nahm, 2 Jan 2025). In Pin(2)-monopole Floer, grading lifts are to $\mathbb{Q}$ and interact with $R$-module actions.

Explicitly, merge/split bands can be interpreted algebraically by working with power series in $U_1$, $U_2^{1/2}$ and using module projections/injections $U_1=U_2$ or $U_2 \to U_1^2$—these encode merge or split operations at the level of link Floer homology (Nahm, 2 Jan 2025).

5. Proof Outline and Detection Lemmas

Proof of exactness employs a reduction to local models:

• Step 1: Localize the argument to a 3-punctured torus Heegaard diagram with attaching curve families $\beta_a,\beta_b,\beta_c \subset T^2$, special generators $\tau, \rho, \sigma$.
• Step 2: Verify maps’ composition properties, showing that composite maps are chain homotopic to particular top-degree cycles.
• Step 3: Invoke the mapping-cone/triangle-detection lemma (twisted-complex language) to guarantee exactness at all vertices.
• Step 4: Extend results to arbitrary Heegaard triples via stabilization and handleslides, preserving grading uniqueness for special cycles.

In Pin(2)-monopole Floer, the iterated mapping cone and pentagon-of-metrics arguments provide functoriality and uniqueness for the non-cobordism map, enforcing the triangle (spectral sequence analysis and Morse–Bott formalism) (Lin, 2015).

6. Applications and Implications for Knot and 3-Manifold Invariants

A salient application involves surgery on alternating knots. For $K \subset S^3$ of signature $\sigma \leq 0$ and Arf invariant $\operatorname{Arf}(K)$, the distance-two triangle enables recovery of Floer groups for $S^3_{\pm1}(K)$ and extraction of Manolescu's correction terms $(\alpha, \beta, \gamma)$ from the module decomposition, following Ozsváth–Szabó's computation for lens space surgeries. Explicit closed-form tables classify the correction terms depending on $\sigma$ and $\operatorname{Arf}(K)$:

• If $\operatorname{Arf}(K) = 0$, all $\alpha, \beta, \gamma$ for $S^3_{-1}(K)$ vanish.
• If $\operatorname{Arf}(K) = 1$, values follow a $4$-fold pattern mod $8$ in $\sigma$.

As a corollary, one can prove certain homology 3-spheres $S^3_{-1}(K)$ are not homology-cobordant to any Seifert fiber space; their correction terms violate necessary inequalities (cf. [(Lin, 2015), Prop. 1.9]). This suggests the triangle serves as a powerful tool for distinguishing 3-manifold invariants post-surgery and for the computation of more refined invariants in various Floer-theoretic settings.