Instanton summer school

Abstracts

Raphael Zentner: mini-course on Instanton gauge theory and applications to 3-manifold topology
The first half of the course will cover foundational material on instantons, including a discussion of the relevant analytical background on elliptic operators; in particular, we will give a proof of Uhlenbeck compactness.
In the second half we will look at some applications, stemming from vanishing/non-vanishing results for Donaldson invariants, instanton Floer homology, and holonomy perturbations; among them, Property P, SU(2)-representations of surgeries and knot complements, splicings, and integral homology spheres, and some new results, obtained jointly with Steven Sivek.

Stefan Behrens: Frøyshov-type invariants via homotopy theory
We will discuss Froyshov's h-invariants in monopole Floer homology from the perspective of Manolescu's Seiberg–Witten–Floer homotopy types and how this interpretation leads to the definition of 'higher h-invariants' which potentially contain more information. The goal of the first talk is to explain the approach while the second will give a gentle introduction to the homotopy theoretic aspects.

Daniele Celoria: Concordance bounds on the Thurston norm
We will show that the Behrens–Golla fully twisted correction terms in Heegaard Floer provide lower bounds on the Thurston norm of certain homology classes of a link complement, up to concordance. We then specialise this procedure to knots in S²×S¹, and obtain a lower bound on their geometric winding number. This is joint work with M. Golla.

Anthony Conway: A multivariable Casson–Lin type invariant
In 1992, X-S. Lin defined a knot invariant via a signed count of traceless irreducible SU(2)-representations of the knot group. While an analogous count of representations had first been performed by Casson for homology 3-spheres, Lin additionally related his invariant to the knot signature. After recalling Lin's construction as well as results of Heusener–Kroll, we discuss a generalisation of this invariant to links of several components. This is joint work with Léo Bénard.

Kyle Larson: Linear independence in the rational homology cobordism group
We will discuss simple homological conditions for a rational homology 3-sphere Y to have infinite order in the rational homology cobordism group, and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when Y is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums. This is joint work with Marco Golla.