# Seminars & Events for Topology Seminar

##### Equivariant maps from a configuration space to a sphere

THIS IS A JOINT TOPOLOGY/ALGEBRAIC TOPOLOGY SEMINAR. There are several distinct reasons to ask for the existence of an S_n-equivariant map from the configuration space F(R^d,n) of n labeled points in R^d to a certain S_n-representation sphere of dimension (d+1)(n-1)-1. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Naturality in sutured monopole homology

Kronheimer and Mrowka defined a version of monopole Floer homology which assigns to any balanced sutured manifold a module up to isomorphism. In this talk, I will discuss how to replace “a module up to isomorphism” with something more natural, by showing that different choices made in the construction are related by canonical isomorphisms which are well-defined up to multiplication by a unit. This allows us to construct some interesting functors out of sutured monopole homology and to talk about elements of it, and I will outline some applications of this fact to contact geometry. This is joint work with John Baldwin.

##### 3-Manifold Mutations Detected by Heegaard Floer Homology

Given a self-diffeomorphism h of a closed, orientable surface S and an embedding f of S into a three manifold M, we construct a mutant manifold N by cutting M along f(S) and regluing by h. We will consider whether there are any gluings such that for any embedding, the manifold and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if the gluing is not isotopic to the identity or the genus 2 hyperelliptic involution, then there exists an embedding of S into a three manifold M such that the rank of the nontorsion summands of the Heegaard Floer homology of M differs from that of its mutant.

##### Length and volume in contact three-manifolds

We give an introduction to a theorem (joint with Dan Cristofaro-Gardiner and Vinicius Ramos) that relates the volume of a contact three-manifold to the lengths of certain collections of closed orbits of the Reeb vector field. This implies that the Reeb vector field always has at least two closed orbits. We also discuss a conjecture on the existence of a Reeb orbit which is short with respect to the volume.

##### Gromov-Uhlenbeck Compactness and the Atiyah-Floer Conjecture

Let M be a symplectic manifold with a hamiltonian group action by G. We introduce an analytic framework that relates holomorphic curves in the symplectic quotient of M to gauge theory on M. As an application of these ideas, we discuss the relation between instanton Floer homology and Lagrangian Floer homology of representation varieties.

##### Plane Floer Homology and Knot Concordance Group

Plane Floer Homology defines a functor from the category of 3-manifolds and cobordisms to the category of vector spaces over an appropriate Novikov field N. Like other Floer homologies assigned to 3-manifolds, this homology theory carries one important property, i.e., surgery exact triangle or more generally ``cubical surgery relation''. As a result, for a classical link we derive a spectral sequence whose second page is a suitable variant of Khovanov homology and it abuts to the plane floer homology of the double cover of S^3 branched along the link. Moreover, plane Floer homology can be easily computed. As an application we will show how to define a family of knot concordance invariants.

##### Homological stability for moduli spaces of manifolds

THIS IS A JOINT TOPOLOGY / ALGEBRAIC TOPOLOGY SEMINAR. There will be two separate talks: 3:00-4:00 pm (Fine 214) and 4:30-5:30 pm (Fine 314). For a compact manifold $W$, possibly with boundary, we shall let $\mathrm{Diff}(W)$ denote the topological group of diffeomorphisms of $W$ fixing a neighborhood of $\partial W$. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Topological Actions of Connected Compact Lie Groups on Manifolds

THIS IS A JOINT TOPOLOGY / ALGEBRAIC TOPOLOGY SEMINAR. We survey some new methods and results on existence and on topological classification of actions of connected, compact Lie groups on manifolds. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Generating the Fukaya category

The Fukaya category is an interesting invariant of a symplectic manifold. It is, at first sight, a rather fearsome thing: its objects are all Lagrangian submanifolds, an enormous and unruly set. Nevertheless, in certain circumstances one can find a finite set of Lagrangians which `generate' the category in an appropriate sense. I will explain a criterion, due to Abouzaid-Fukaya-Oh-Ohta-Ono, for when this happens. I will then explain a result, due to Tim Perutz and myself, which shows that this criterion is satisfied automatically in a large number of cases which arise naturally in the context of homological mirror symmetry. I hope it will inspire audience members with an interest in Heegaard Floer homology to try some computations in the Fukaya category of a symmetric product.

##### An infinite rank summand of topologically slice knots

Let C_{TS} be the subgroup of the smooth knot concordance group generated by topologically slice knots. Endo showed that C_{TS} contains an infinite rank subgroup, and Livingston and Manolescu-Owens showed that C_{TS} contains a Z^3summand. We show that in fact C_{TS} contains a Z^\infty summand. The proof relies on the knot Floer homology package of Ozsvath-Szabo and the concordance invariant epsilon.

##### Virtual domination of $3$-manifolds

For any closed oriented hyperbolic $3$-manifold $M$, and any closed oriented $3$-manifold $N$, we will show that $M$ admits a finite cover $M'$, such that there exists a degree-$2$ map $f: M' \rightarrow N$.

##### Lattice cohomology

Any negative definite plumbed 3-manifold has its lattice cohomology, determined from the lattice of one of its plumbing representations. I will present several properties of the these modules, e.g., the `reduction theorem', which reduced the rank of the lattice to the number of `bad vertices'. Furthermore, I will define a modified version of the theory, called `path lattice cohomology'. I will discuss its motivation and connection with the theory of surface singularities.

##### Totally disconnected groups (not) acting on three-manifolds

Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery--Zippin. A stronger conjecture

is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a

corollary to the work of Gleason and Montgomery--Zippin) that it suffices to rule out the case of the additive group of $p$-adic integers acting faithfully on a manifold. I will present a solution in dimension three.

The proof uses tools from low-dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group.

##### Non-displaceable Lagrangians via minimal model transitions

**This is a joint Topology/Symplectic Geometry seminar. **I will discuss some results, some older and some newer, on the general idea that in many birationally-Fano cases, generators of the Fukaya category seem to arise from transitions in the minimal model program. A specific result from a couple of years ago, joint with Gonzalez, is that the number of non-displaceable Lagrangian tori in a smooth projective toric variety is at least the number of transitions in a toric minimal model program. A newer specific result is that the blow-up of a smooth projective Fano variety at a finite set contains non-displaceable Lagrangian tori of number at least the order of the set, for sufficiently small exceptional divisor.

##### Knot Floer homology and the unoriented 4-ball genus

##### Bordered Heegaard Floer homology and 4-manifolds with corners

Lipshitz, Ozsváth and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism to each such cobordism between two mapping cylinders. Like the Heegaard Floer invariants associated to cobordisms between closed 3-manifolds, this morphism arises from counting holomorphic triangles on Heegaard triples. We demonstrate that the homotopy class of the induced morphism only depends on the symplectic structure of the cobordism in question.

##### Transverse invariants in filtrations of Khovanov homology

In 2005, O. Plamenevskaya used Khovanov homology to define an invariant \psi of transverse knots T in R^3. Since then, related invariants have been defined using spectral sequences that start at Khovanov homology. On a similar note, Baldwin and Plamenevskaya related \psi to the Heegaard Floer contact invariant of the double branched cover of T using the Ozsvath-Szabo spectral sequence. However, it has been unknown whether \psi was a strictly stronger invariant of T than the classical self-linking number. I will give a proof that \psi is in fact determined by self-linking, and explain how a similar argument shows that the contact invariant of the double branched cover is also determined by self-linking.

##### Homology three-spheres and surgery obstructions

The Lickorish-Wallace theorem states that every closed, connected, orientable three-manifold can be expressed as surgery on a link in the three-sphere. It is natural to ask which three-manifolds can be obtained by surgery on a knot in the three-sphere. We discuss a new way to obstruct integer homology spheres from being surgery on a knot and give some examples. This is joint work with Jennifer Hom and Cagri Karakurt.

##### Non-simple genus minimizers in lens spaces

When does a knot in a 3-manifold have the smallest genus amongst all knots homologous to it? This is a basic but largely unexplored problem, and addressing it for lens spaces is closely connected with the Berge conjecture, which posits when surgery along a knot in the three-sphere can produce a lens space. I will describe this connection in greater detail and then exhibit some examples of non-simple genus-minimizing knots in lens spaces. This is joint work with Yi Ni.

##### Elliptic Curves and Knot Homology

Given a smooth elliptic curve E over the complex numbers we construct a functor-valued invariant of tangles, extending a known braid group action on the derived category of coherent sheaves on E^n. The invariant associated to a closed link L is related to odd Khovanov homology, and can be described in terms of the double cover branched over L.