Surgery, Polygons and SU(N)-Floer Homology

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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 Aₙ. 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.