Equivariant aspects of singular instanton Floer homology

Daemi, Aliakbar¹ ; Scaduto, Christopher²

¹ Department of Mathematics and Statistics, Washington University in St Louis, St Louis, MO, United States
² Department of Mathematics, University of Miami, Coral Gables, FL, United States

Geometry & topology, Tome 28 (2024) no. 9, pp. 4057-4190.

Résumé

We associate several invariants to a knot in an integer homology $3$–sphere using $\mathrm{SU}(2)$ singular instanton gauge theory. There is a space of framed singular connections for such a knot, equipped with a circle action and an equivariant Chern–Simons functional, and our constructions are morally derived from the associated equivariant Morse chain complexes. In particular, we construct a triad of groups analogous to the knot Floer homology package in Heegaard Floer homology, several Frøyshov-type invariants which are concordance invariants, and more. The behavior of our constructions under connected sums are determined. We recover most of Kronheimer and Mrowka’s singular instanton homology constructions from our invariants. Finally, the ADHM description of the moduli space of instantons on the $4$–sphere can be used to give a concrete characterization of the moduli spaces involved in the invariants of spherical knots, and we demonstrate this point in several examples.

Détail

DOI : 10.2140/gt.2024.28.4057

Classification : 57R58, 57M27
Keywords: singular instantons, equivariant Floer homology, knot concordance

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