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

Lucas Culler; Aliakbar Daemi; Yi Xie

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arXiv · 1712.09910

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

Lucas Culler·Aliakbar Daemi·Yi Xie

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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.

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BibTeX RIS

Lucas Culler, Aliakbar Daemi, Yi Xie. 2017-12-28. Surgery, Polygons and $SU(N)$-Floer Homology. https://arxiv.org/abs/1712.09910

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