[Paper Review] Exact triangles in Seiberg-Witten Floer theory. Part III: proof of exactness

Matilde Marcolli, Bai‐Ling Wang|ArXiv.org|Sep 15, 2000

Geometric and Algebraic Topology21 references3 citations

TL;DR

This paper establishes the exactness of a triangle in Seiberg-Witten Floer homology associated with knot surgery on a homology 3-sphere, using geometric limits and gluing techniques on punctured cobordisms. It proves that the sequence $ C_*(Y_1) \xrightarrow{w^1_*} C_*(Y,\mu) \xrightarrow{w^0_*} \bigoplus_k C_{(*)}(Y_0,\mathfrak{s}_k) \xrightarrow{\bar{w}^2_*} C_*(Y_1)[-1] $ forms a distinguished triangle in the derived category of chain complexes, confirming the existence of a canonical exact triangle in the context of Seiberg-Witten Floer theory.

ABSTRACT

This is the third part of the work on the exact triangles. We construct chain homomorphisms and show exactness of the resulting sequence.

Motivation & Objective

• To establish the exactness of a triangle in Seiberg-Witten Floer homology arising from 0- and 1-surgery on a knot in a homology 3-sphere.
• To construct chain maps $ w^1_* $ and $ w^0_* $ via surgery cobordisms $ W_1 $ and $ W_0 $, compatible with the moduli space decomposition.
• To prove that the composition $ w^0_* \circ w^1_* = 0 $, and that $ w^0_* $ is surjective and $ w^1_* $ is injective, ensuring exactness in the middle term.
• To show that the resulting sequence forms a distinguished triangle in the derived category of chain complexes, using geometric limits and gluing of finite energy monopoles.

Proposed method

• Constructs chain maps $ w^1_* $ and $ w^0_* $ using spin^c structures on surgery cobordisms $ W_1 $ and $ W_0 $ that restrict to given structures on the boundary components.
• Analyzes geometric limits of moduli spaces of finite energy monopoles on $ V \times \mathbb{R} $, where $ V $ is the knot complement.
• Uses pre-gluing data from holomorphic triangles in a covering of the character variety $ \chi(T^2) $, with boundaries along Lagrangians determined by the spin^c structures.
• Applies a gluing theorem to identify moduli spaces on punctured cobordisms with pre-gluing data, establishing diffeomorphisms between moduli spaces on $ W_1 $ and $ W_0 $.
• Compares zero-dimensional moduli spaces on $ W_1 $ and $ W_0 $ to show an orientation-reversing diffeomorphism, proving $ w^0_* \circ w^1_* = 0 $.
• Uses the convergence of geometric limits as $ \epsilon \to 0 $ to equate algebraic counts of solutions in the moduli space on the composite cobordism with those in the original Floer complex.

Experimental results

Research questions

• RQ1 Does the sequence of chain complexes induced by 0- and 1-surgery on a knot in a homology 3-sphere form an exact triangle in Seiberg-Witten Floer homology?
• RQ2 Can the maps $ w^1_* $ and $ w^0_* $, defined via surgery cobordisms, be shown to satisfy the exactness condition $ w^0_* \circ w^1_* = 0 $?
• RQ3 How do geometric limits of finite energy monopoles on $ V \times \mathbb{R} $ and holomorphic triangles in $ \chi(T^2) $ contribute to the gluing of solutions on the cobordisms?
• RQ4 Is the resulting sequence quasi-isomorphic to a standard distinguished triangle in the derived category of chain complexes?
• RQ5 Can the algebraic counts of flow lines in the moduli space on the composite cobordism be matched to those in the original Floer complex via geometric limits?

Key findings

• The composition $ w^0_* \circ w^1_* = 0 $ is proven via an orientation-reversing diffeomorphism between moduli spaces $ \mathcal{M}^{W_1}_\ell(a_1,a) $ and $ \mathcal{M}^{W_0}_k(j(a_1),\pi(a)) $, implying the exactness condition.
• The map $ w^1_* $ is injective and $ w^0_* $ is surjective, as shown by the pairing identities $ \langle j(a_1'), w^1_*(a_1) \rangle = \delta_{a_1',a_1} $ and $ \langle a_0, w^0_*(a) \rangle = \delta_{a_0,\pi(a)} $.
• The algebraic count of solutions in the zero-dimensional moduli space $ \mathcal{M}^{\bar{W}_2}_k(a_j^{(0)}, a_i^{(1)}) $ agrees with the count in the original complex: $ N^{\bar{W}_2}_k(a_j^{(0)}, a_i^{(1)}) = n_{Y,\mu}(a_j^{(0)}(\epsilon), a_i^{(1)}(\epsilon)) $.
• For sufficiently small $ \epsilon \leq \epsilon_0 $, the sequence $ C_*(Y_1) \xrightarrow{w^1_*} C_*(Y,\mu) \xrightarrow{w^0_*} \bigoplus_k C_{(*)}(Y_0,\mathfrak{s}_k) \to 0 $ becomes an exact triple of complexes.
• The full sequence $ C_*(Y_1) \xrightarrow{w^1_*} C_*(Y,\mu) \xrightarrow{w^0_*} \bigoplus_k C_{(*)}(Y_0,\mathfrak{s}_k) \xrightarrow{\bar{w}^2_*} C_*(Y_1)[-1] $ is a distinguished triangle in the derived category of chain complexes.
• The geometric limits of solutions on $ W_1 \setminus \{x_1\} $ and $ W_0 \setminus \{x_0\} $, after stretching product regions, converge to solutions on the composite cobordism $ W = W_1 \cup_Y W_0 $, enabling the gluing and counting arguments.