Surgery, Polygons and $SU(N)$-Floer Homology

TL;DR

This paper explores how $SU(N)$-instanton Floer homology behaves under Dehn surgery, establishing new exact polygon relations for $SU(3)$ and $SU(4)$, and conjecturing a general pattern for all $N$.

Contribution

It introduces surgery exact tetragons and pentagons for $SU(3)$ and $SU(4)$-Floer homologies and conjectures a universal $(N+1)$-gon structure for all $SU(N)$ cases.

Findings

Established surgery exact tetragons for $SU(3)$-Floer homology.

Established surgery exact pentagons for $SU(4)$-Floer homology.

Conjectured the existence of surgery exact $(N+1)$-gons for general $SU(N)$.

Abstract

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 $SU(N)$-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 $SU(3)$- and $SU(4)$-instanton Floer homologies. It is also conjectured that $SU(N)$-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.