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Surgery, polygons and SU ( N ) ‐Floer homology
Author(s) -
Culler Lucas,
Daemi Aliakbar,
Xie Yi
Publication year - 2020
Publication title -
journal of topology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.447
H-Index - 31
eISSN - 1753-8424
pISSN - 1753-8416
DOI - 10.1112/topo.12137
Subject(s) - floer homology , mathematics , dehn surgery , instanton , homology (biology) , combinatorics , morse homology , torus , knot (papermaking) , pure mathematics , curvature , topology (electrical circuits) , cellular homology , mathematical physics , geometry , biochemistry , chemistry , symplectic geometry , chemical engineering , engineering , gene
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.