Knot surgery formulae for instanton Floer homology, I: The main theorem

Li, Zhenkun¹ ; Ye, Fan²

¹Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
²Department of Mathematics, Harvard University, Cambridge, MA, United States

Geometry & topology, Tome 29 (2025) no. 5, pp. 2269-2342

Résumé

We prove an integral surgery formula for framed instanton homology I♯(Y m(K)) for any knot K in a 3-manifold Y with [K] = 0 ∈ H1(Y ; ℚ) and m≠0. Although the statement is similar to Ozsváth–Szabó’s integral surgery formula for Heegaard Floer homology, the proof is new and based on sutured instanton homology SHI ⁡ and the octahedral lemma in the derived category. As byproducts, we obtain a formula computing instanton knot homology of the dual knot analogous to Eftekhary’s and Hedden–Levine’s work, and also an exact triangle between I♯(Y m(K)), I♯(Y m+k(K)) and k copies of I♯(Y ) for any m≠0 and large k. In the proof of the formula, we discover many new exact triangles for sutured instanton homology and relate some surgery cobordism map to the sum of bypass maps, which are of independent interest. In a companion paper, we derive many applications and computations based on the integral surgery formula.

Détail

DOI : 10.2140/gt.2025.29.2269

Keywords: instanton, Floer homology, Dehn surgery, mapping cone formula

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