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From Higher Invariants

# **Conference Gauge Theory and Applications -- Abstracts **

## Kim Froyshov

- Title: Instanton Floer homology over Z/2Z

- Abstract: One of the main challenges when constructing an instanton Floer theory is how to incorporate the reducible flat connections in such a way as to get Floer groups with good functoriality properties with respect to cobordisms. I will outline one way to solve this problem for mod 2 instanton homology of integral homology 3-spheres. The resulting theory has some formal analogies with Heegaard and monopole Floer theory. This is a report on work in progress.

## Marco Golla

- Title: Rational cuspidal curves and symplectic fillings

- Abstract: A rational cuspidal curve is a complex curve whose singularities are irreducible (i.e. they have connected link), and that are homeomorphic to a 2-sphere. These are more elusive objects than one could expect, and, for instance, their classification in the projective plane is not yet complete. I will discuss a symplectic perspective on the problem and some results obtained with Laura Starkston (in progress).

## Doris Hein

- Title: Transversality in local Morse Theory

- Abstract: The S^1-symmetry in closed orbits of autonomous Hamiltonian systems and Reeb flows presents a difficulty in estimating their number and developing useful Floer-type theories. Using Morse theory, this symmetry can locally be reduced to the study of a finite cyclic group action. In this talk, I will explain how transversality can be achieved for local Morse homology with such symmetries, which is not possible in the global case. This transversality result can be used to define an invariant for periodic orbits which can still be studied by looking at the finite symmetries.

## Tsuyoshi Kato

- Title: Twisted Donaldson invariant

- Abstract: We define a twisted Donaldson’s invariant using the family of Dirac operators twisted by flat bundles, and verify that our formula is non trivial by particular examples. We follow Lusztig and then Connes-Moscovici’s methods to Novikov conjecture by using cyclic cohomology. This leads to a natural question how a classical result in Donaldson theory is also satisfied for the invariant in the case of non admissible fundamental groups.

## Hokuto Konno

- Title: Gauge theory for families of 4-manifolds

- Abstract: Using SO(3)-Yang-Mills theory and Seiberg-Witten theory, I will construct characteristic classes of 4-manifold bundles. These characteristic classes are extensions of the SO(3)-Donaldson invariant and the Seiberg-Witten invariant to families of 4-manifolds, and can detect non-triviality of smooth 4-manifold bundles. The basic idea of the construction of these characteristic classes is to consider an infinite dimensional analogue of classical characteristic classes of manifold bundles, typified by the Mumford-Morita-Miller classes for surface bundles.

## Francesco Lin

- Title: The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds

- Abstract: While both hyperbolic geometry and Floer homology have both been tremendously successful tools when studying three-dimensional topology, their relationship is still very mysterious. In this talk, we provide sufficient conditions for a hyperbolic rational homology sphere not to admit irreducible solutions to the Seiberg-Witten equations in terms of its volume and the length spectrum (i.e. the set of lengths of closed geodesics). We discuss several explicit examples in which this criterion can be applied. This is joint work in progress with Michael Lipnowski.

## Jianfeng Lin

- Title: On monopole Lefschetz number

- Abstract: Monopole Floer homology is a powerful invariant of in 3-dimensional topology. By its naturality, monopole Floer homology of a 3-manifold is acted upon by its mapping class group. In a recent work with Danny Ruberman and Nikolai Saveliev, we make the first step towards understanding this action by calculating the Lefschetz numbers of certain involutions making the 3–manifold into a double branched cover over a link in the 3–sphere. Various applications of of this formula will be discussed.

## Shinichiroh Matsuo

- Title: tba

- Abstract: tba.

## Catherine Meusburger

- Title: Mapping class group actions in Hopf algebra gauge theory

- Abstract: Group valued latttice gauge theories can be generalised to lattice gauge theories on a ribon graph with values in finite-dimensional ribbon Hopf algebra. The resulting model is equivalent to the quantum moduli algebra from the combinatorial quantisation of Chern-Simons theory and, in the case of Drinfeld doubles of semisimple Hopf algebras, to Kitaev lattice models.

We derive a simple description of the mapping class group actions in these models that describes Dehn twists as as well as the mapping class group action on trivalent graphs via flip moves.

## S. Michael Miller

- Title: Equivariant instanton homology and group cohomology

- Abstract: Floer's celebrated instanton homology groups are defined for integer homology spheres, but analagous groups in Heegaard Floer and Monopole Floer homology theories are defined for all 3-manifolds; these latter groups furthermore come in four flavors, and carry extra algebraic structure. Any attempt to extend instanton homology to a larger class of 3-manifolds must be somehow equivariant - respecting a certain SO(3)-action. We explain how ideas from group cohomology and algebraic topology allow us to define four flavors of instanton homology for rational homology spheres, and how these invariants relate to existing instanton homology theories.

## Tomasz Mrowka

- Title: tba

- Abstract: tba

## Jacob Rasmussen

- Title: SL_2(R) representations of knot complements and an extended Lin invariant

- Abstract: Let K be a knot in S^3. In the early 90's, X.S. Lin introduced a Casson-type invariant of K by counting SU(2) representations of the complement of K with fixed holonomy around the meridian. This invariant was later shown to be equivalent to the Levine-Tristam signature of K. I'll discuss a similar invariant which is defined using SL_2(R) in place of SU(2) and does not seem to be determined by known invariants. I'll give some applications to the problem of finding left-orderings on Dehn fillings and branched covers of K, and state some conjectures about the invariant. This is work in progress with Nathan Dunfield.

## Sarah Rasmussen

- Title: tba

- Abstract: tba.

## Daniel Ruberman

- Title: The splitting formula for LSW and applications

- Abstract: I will review the definition of the invariant LSW(X) for manifolds with the homology of S^1 x S^3. It is a combination of a signed count of irreducible solutions to the Seiberg-Witten equations on X with an index-theoretic correction term. Recent work with Jianfeng Lin and Nikolai Saveliev gives a splitting formula for LSW(X) in the case that X contains a rational homology sphere (carrying the 3-dimensional homology). I will give some applications of this splitting formula to the existence of positive scalar curvature metrics, homology cobordism, and knot concordance.

## Dietmar Salamon

- Title: An open-closed isomorphism in instanton Floer homology

- Abstract: tba.

## Nikolai Saveliev

- Title: Link homology and equivariant gauge theory

- Abstract: The talk concerns the singular Floer homology of knots and links defined by Kronheimer and Mrowka using gauge theory on orbifolds; it has been instrumental in proving that the Khovanov homology is an unknot detector. We show how replacing gauge theory on an orbifold with equivariant gauge theory on its double branched cover simplifies matters and allows for explicit calculations for several classes of knots and links. This is a joint work with Prayat Poudel.

## Christopher Scaduto

- Title: Yang-Mills theory and definite intersection forms bounded by homology 3-spheres

- Abstract: Using Yang-Mills instanton Floer theory, we find new constraints on the possible definite intersection forms of smooth 4-manifolds bounded by integer homology 3-spheres. We will give examples of 3-manifolds Y such that the set of definite lattices arising from 4-manifolds with boundary Y consists of essentially two distinct non-standard lattices. The methods used follow the work of Froyshov.

## Steven Sivek

- Title: SU(2)-cyclic surgeries and applications

- Abstract: A rational homology 3-sphere is called SU(2)-cyclic if its fundamental group only admits representations to SU(2) with cyclic image. In the first part of this talk, we will see that for any fixed nontrivial knot K in S^3, the set of SU(2)-cyclic Dehn surgery slopes for K is bounded. In the second part, we will discuss examples of SU(2)-cyclic surgeries and use them to exhibit infinitely many 3-manifolds which have weight one fundamental group but do not arise as Dehn surgery on any knot in S^3. Both parts are joint work with Raphael Zentner.

## Matthew Stoffregen

- Title: Divisibility in the Homology Cobordism Group

- Abstract:We apply Hendricks-Manolescu's involutive Heegaard Floer homology to study the homology cobordism group. In this talk, we will introduce some computable invariants coming from involutive Heegaard Floer homology that obstruct divisibility of classes in the homology cobordism group. As an application, we see that many Seifert-fibered integral homology spheres are not multiples of any class in the homology cobordism group. This is joint work with Irving Dai, Jen Hom, and Linh Truong.

## Speakers

The preliminary list of speakers is:

**Kim Froyshov**(Oslo)**Marco Golla**(Nantes)**Doris Hein**(Freiburg)**Tsuyoshi Kato**(Kyoto)**Hokuto Konno**(Univerisity of Tokyo)**Francesco Lin**(Princeton)**Jianfeng Lin**(MIT)**Shinichiroh Matsuo**(Nagoya)**Catherine Meusburger**(Erlangen-Nürnberg)**Michael Miller**(UCLA)**Tomasz Mrowka**(MIT)**Jacob Rasmussen**(Cambridge)**Sarah Rasmussen**(Cambridge)**Daniel Ruberman**(Brandeis University)**Dietmar Salamon**(ETH Zürich)**Nikolai Saveliev**(University of Miami)**Christopher Scaduto**(Stony Brook)**Steven Sivek**(Imperial College London)**Matthew Stoffregen**(MIT)