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.

中文翻译：

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手术，多边形和SU（N）-Floer同源性

各种3流形Floer同源性理论中的手术精确三角形为研究和计算相关Floer同源性组提供了重要工具。这些确切的三角形与3个流形的不变量相关，这是由三个在固定结上的不同Dehn手术获得的。本文的行为$\mathrm{苏}（ñ）$研究了Dehn手术的Instanton Floer同源性。特别是，显示出分别有手术精确的四边形和五边形$\mathrm{苏}（3）$-和 $\mathrm{苏}（4）$-Instanton Floer同源性。也有人推测$\mathrm{苏}（ñ）$-Instanton Floer同源性通常承认手术准确 $（ñ+\mathrm{1个}）$-gon。证明中必不可少的步骤是在ALE型空间上构造一个渐近圆柱度量族${\mathrm{一种}}_{ñ}$。这个家庭被$（ñ-2）$维关联体，由标量曲率为正的反自对偶度量组成。家庭中的指标也承认环面对称。