Lucas Culler, Aliakbar Daemi, Yi Xie

Surgery exact triangles in various 3‐manifold Floer homology theories provide an important tool in studying and computing the relevant Floer homology groups. These exact triangles relate the invariants of 3‐manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, the behavior of $\mathrm{SU}(N)$‐instanton Floer homology with respect to Dehn surgery is studied. In particular, it is shown that there are surgery exact tetragons and pentagons, respectively, for $\mathrm{SU}(3)$‐ and $\mathrm{SU}(4)$‐instanton Floer homologies. It is also conjectured that $\mathrm{SU}(N)$‐instanton Floer homology in general admits a surgery exact $(N+1)$‐gon. An essential step in the proof is the construction of a family of asymptotically cylindrical metrics on ALE spaces of type ${A}_{n}$. This family is parametrized by the $(n-2)$‐dimensional associahedron and consists of anti‐self‐dual metrics with positive scalar curvature. The metrics in the family also admit a torus symmetry.