# Seminars & Events for Topology Seminar

##### Homology cobordism and surgery on alternating knots

We discuss a few computations in Pin(2)-monopole Floer homology, and their applications. Our main protagonists are homology spheres obtained by surgery on alternating knots.

##### Conway mutation and knot Floer homology

Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in Z/2Z is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that delta-graded knot Floer homology, a singly-graded reduction of the full invariant, is preserved by mutation. In this talk, I will give a new proof that Khovanov homology mod 2 is mutation-invariant. The same strategy can be applied to delta-graded knot Floer homology and proves the Baldwin-Levine conjecture for mutations on a large class of tangles.

##### Stein fillability in higher dimensions

We recall the definition of the surgery obstruction for an almost contact manifold to be Stein fillable. This class can be used to prove various fillability and non-fillability results. This is joint work with J. Bowden and D. Crowley.

##### Co-oriented Taut Foliations

I will describe a new construction of (codimension one) co-oriented taut foliations (CTFs) of 3-manifolds. It follows from this construction that if K is either an alternating knot or a Montesinos knot, then K is not an L-space knot if and only if every nontrivial Dehn filling on K yields a 3-manifold that contains a CTF. This work is joint with Charles Delman.

##### Bordered Floer homology via immersed curves II

I'll describe parts of an ongoing project with Jonathan Hanselman and Jake Rasmussen, which interprets bordered Floer homology for manifolds with torus boundary in terms of systems of immersed curves in the punctured torus.

##### Higher Order Corks

In 1960, Mazur constructed a contractible 4-manifold W with non-simply connected boundary whose product with an interval is the 5-ball. Thirty years later, Akbulut showed that ∂W supports an involution T that does not extend to a diffeomorphism of W, thus producing the first nontrivial *involutory cork *(W,T). Akbulut's proof was to embed W in a smooth 4-manifold so that the resulting *cork twist* (cut out W and reglue using T) changes the ambient smooth structure. It is now known by the *involutory cork theorem *of Curtis-Freedman-Hsiang-Stong and Matveyev that any two smooth structures on a closed, simply connected 4-manifold are related by a single involutory cork twist. The existence of higher order corks (where T is of order >2) whose twists by powers of T can produce distinct smooth structures was unknown before this year

##### Cobordism maps in link Floer homology

Given a (decorated) link cobordism between two links K and L (that is, an embedded surface in S^3 x [0,1] that K and L co-bound), Juhász defined a map between their link Floer homologies. We prove that when the surface is an annulus the map preserves the natural bigrading of HFL and is always non-zero. This has some interesting applications, in particular the existence of a non-zero element in HFL(K) associated to each properly embedded disc in B^4 whose boundary is the knot K in S^3. I will then discuss some properties of the map in HFL when the cobordism is not necessarily given by an annulus. This is joint work with András Juhász.

##### A monopole invariant for foliations without transverse invariant measure

The question about existence and flexibility of taut foliations on a three manifold has been studied for decades. Floer-theoretical obstructions for the existence of taut foliations on rational homology spheres have been obtained by Kronheimer, Mrowka, Ozsvath, and Szabo by perturbing of the foliation to contact structures. Recently, by showing that the perturbed contact structure is unique in many cases, Vogel and Bowden constructed examples of taut foliations that are homotopic as distributions but can not be deformed to each other through taut foliations. In this talk we will propose a different approach. Instead of perturbing the foliation to a contact structure, we directly study a symplectization of the foliation itself, and that leads to a canonically defined class in the monopole Floer homology.

##### TBA - Tolga Etgü

##### Heegaard Floer invariants for homology S^1 \times S^3s

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of S^1 \times S^3. Specifically, we show that for any smoothly embedded 3-manifold Y representing a generator of H_3(X), a suitable version of the Heegaard Floer d invariant of Y, defined using twisted coefficients, is a diffeomorphism invariant of X. We show how this invariant can be used to obstruct embeddings of certain types of 3-manifolds, including those obtained as a connected sum of a rational homology 3-sphere and any number of copies of S^1 \times S^2. We also give similar obstructions to embeddings in certain open 4-manifolds, including exotic R^4s. This is joint work with Danny Ruberman.

##### Pin(2)-equivariant Floer homology and homology cobordism

We review Manolescu's construction of the Pin(2)-equivariant Seiberg-Witten Floer stable homotopy type, and apply it to the study of the 3-dimensional homology cobordism group. We introduce the 'local equivalence' group, and construct a homomorphism from the homology cobordism group to the local equivalence group. We then apply Manolescu's Floer homotopy type to obstruct cobordisms between Seifert 0 spaces. In particular, we show the existence of integral homology spheres not homology cobordant to any Seifert space. We also introduce connected Floer homology, an invariant of homology cobordism taking values in isomorphism classes of modules.

##### The framed instanton homology of a surface times a circle

We compute a version of SO(3) instanton homology for a surface times a circle with non-trivial bundle. This is roughly the Morse homology of a functional whose critical set is a space of framed flat SO(3)-connections on a surface.

##### On Thurston’s Euler class one conjecture

In 1976, Thurston proved that taut foliations on closed hyperbolic 3–manifolds have Euler class of norm at most one, and conjectured that, conversely, any Euler class with norm equal to one is Euler class of a taut foliation. I construct counterexamples to this conjecture and suggest an alternative conjecture.

##### Knot traces and concordance

A conjecture of Akbulut and Kirby from 1978 states that the concordance class of a knot is determined by its 0-surgery. In 2015, Yasui disproved this conjecture by providing pairs of knots which have the same 0-surgeries yet which can be distinguished in (smooth) concordance by an invariant associated to the four-dimensional traces of such a surgery. In this talk, I will discuss joint work with Lisa Piccirillo in which we construct many pairs of knots which have diffeomorphic 0-surgery traces yet some of which can be distinguished in smooth concordance by the Heegaard Floer d-invariants of their double branched covers. If time permits, I will also discuss the applicability of this work to the existence of interesting invertible satellite maps on the smooth concordance group.

##### Embedding problems in affine algebraic geometry and slice knots

We first discuss classical questions about polynomial embeddings of the complex line C into complex spaces such as C^m and affine algebraic groups. Next, we consider knots and different notions of sliceness for knots.

Finally, we use a knot theory perspective to indicate proofs for the embedding questions discussed first.

##### Relatively hyperbolic groups vs 3-manifold groups

An illustrative example of a relatively hyperbolic group is the fundamental group of a hyperbolic knot complement. In this case, the peripheral subgroup corresponds to the group of the cusp cross-section, $\mathbb{Z} \oplus \mathbb{Z}$. Bowditch described the boundary of a relatively hyperbolic group pair $(G,P)$ as the boundary of any hyperbolic space that $G$ acts geometrically finitely upon, where the maximal parabolic subgroups are conjugates of the peripheral group $P$. For example, the fundamental group of a hyperbolic knot complement acts as a geometrically finitely on $\mathbb{H}^3$, where the maximal parabolic subgroups are the conjugates of $\mathbb{Z} \oplus \mathbb{Z}$ and its Bowditch boundary is $S^2$.

##### Free Seifert fibered pieces of pseudo-Anosov flows

We prove a structure theorem for pseudo-Anosov flows restricted to Seifert fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration so that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A non periodic Seifert fibered piece is called free. In this talk we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow is orbitally equivalent to a hyperbolic blow up of a geodesic flow piece. A geodesic flow piece is a finite cover of the geodesic flow on a compact hyperbolic surface, usually with boundary (a union of geodesics). The proof uses an associated convergence group theorem, hyperbolic blow ups and models of geodesic flows. This is joint work with Thierry Barbot.

##### Topology of the space of metrics of positive scalar curvature

**This is a joint Algebraic Topology / Topology seminar. **We use recent results on the moduli spaces of manifolds, relevant index and surgery theory to study the index-difference map from the space ${\mathcal R}iem^+(W^d)$ of psc-metrics to the space $\Omega^{d+1}KO$ representing the real $K$-theory. In particular, we show that the index map induces nontrivial homomorphism in homotopy groups $\pi_k {\mathcal R}iem^+(W^d) \to \pi_k \Omega^{d+1}KO$ once the target groups $\pi_k \Omega^{d+1}KO= KO_{k+d+1}$ are not trivial. This work is joint with J. Ebert and O. Randall-Williams. In this talk, I also plan to discuss some recent applications of those results.

##### Stable moduli space of high-dimensional handlebodies

**This is a joint Topology / Algebraic Topology seminar. **We study the moduli space of handlebodies diffeomorphic to $(D^{n+1}\times S^n)^{\natural g}$, i.e. the classifying space $\mathrm{BDiff} (D^{n+1}\times S^n)^{\natural g}, D^{2n})$ of the group of diffeomorphisms that restrict to the identity near an embedded disk $D^{2n} \subset \partial (D^{n+1}\times S^n)^{\natural g}$. We prove that there is a natural map $$\colim_{g\to\infty}\mathrm{BDiff}((D^{n+1}\times S^n)^{\natural g}, D^{2n}) \;\longrightarrow \; Q_{0}BO(2n+1)\langle n \rangle_{+}$$ which induces an isomorphism in integral homology when $n\geq 4$. Above, $BO(2n+1)\langle n \rangle$ denotes the $n$-connective cover of $BO(2n+1)$. This work is joint with N.Perlmutter. I also plan to discuss some recent results related to this work.