UB Geometry and Topology Seminar 

Abstract: A few years ago, Taubes showed that Hutchings's embedded contact homology (ECH) is canonically isomorphic to a version of Seiberg--Witten Floer cohomology. It follows as a consequence that ECH is a topological invariant of the underlying 3-manifold. More recently, Hutchings and Taubes showed that filtered ECH is only dependent on the choice of a contact form. In joint work with Steven Sivek, we prove an analog of the latter result for sutured ECH, defined by Colin, Ghiggini, Honda, and Hutchings for contact 3-manifolds with convex boundary, which are naturally sutured manifolds. Furthermore, we show that sutured ECH is a natural invariant, and that it admits a contact class that is preserved under maps induced by deformations of the contact structure relative to the boundary of the 3-manifold. The aim of this talk is to explain the mechanics of our proof, which uses a compactness result for solutions of the Seiberg--Witten equations on certain non-compact contact 3-manifolds and their symplectizations due to Taubes.

Abstract: We will survey some results concerning the embedding of right-angled Artin groups in diffeomorphism groups of various manifolds, concentrating in dimensions two and one. We will connect these results to some open questions in the theory of braid groups and mapping class groups. This represents joint work with S. Kim.

Abstract: When studying deformations of an $A$-module $M$, Laudal and Yau showed that one can consider $1$-cocycles in the Hochschild cohomology of $A$ with coefficients in the bi-module $End_k(M).$ With this in mind, the use of higher order Hochschild (co)homology, presented by Pirashvili and Anderson, to study deformations seems only natural though the current definition allows only symmetric bi-module coefficients. In this talk we present an extended definition for higher order Hochschild cohomology which allows multi-module coefficients (when the simplicial sets $ X_{\bullet}$ are accommodating) which agrees with the current definition. Furthermore we determine the types of modules that can be used as coefficients for the Hochschild cochain complexes based on the simplicial sets they are associated to.

Abstract: In 1981 Masur proved the existence of a dense Teichmueller geodesic in moduli space. As some form of analogue, we construct dense geodesic rays in certain subcomplexes of the $Out(F_r)$ quotient of outer space. This is joint work in progress with Yael Algom-Kfir.

Abstract: We study some properties of decomposable exact Lagrangian cobordisms between Legendrian links in $\mathbb{R}^3$ with the standard contact structure. In particular, for any decomposable exact Lagrangian filling $L$ of a Legendrian link $K$, we may obtain a normal ruling of $K$ which is associated with $L$. We prove that the associated normal rulings must have even number of clasps. As a result, a Legendrian $(4,-(2n+5))$ -torus knot, $n\geq 0$, does not have a decomposable exact Lagrangian filling because it has only one normal ruling and this normal ruling has odd number of clasps.

Abstract: Let $S_g$ be a closed surface of genus $g$, and let $(\alpha,\beta)$ be a filling pair of essential simple closed curves on $S_g$. It is easy to show that $\alpha,\beta$ must intersect at least $2g-1$ times. What is not so easy to show is that this minimum is realized for all genera $g\neq 2$. We describe a procedure for constructing closed surfaces with minimally intersecting filling pairs via connected sums, and a criterion for determining when such a surface can be decomposed as a connected sum.

Abstract: The triangulation conjecture claimed that any manifold can be triangulated. I will sketch the disproof of the conjecture in dimensions at least $5$. This uses the TQFT properties of the Seiberg-Witten equations, as well as their $Pin(2)$ symmetry.

Abstract: It is well known that any 3-manifold can be obtained by Dehn surgery on a link in the 3-sphere but not which ones can be obtained from a knot or which knots can produce them. We'll talk about using the Heegaard Floer ''correction terms'' or ''d-invariants'' to investigate these two questions for elliptic Seifert fibered spaces (other than lens spaces). We will see a family of examples where explicit calculations of the correction terms completely answer these questions; for a generic elliptic space, we will discuss how the invariants place a bound on the possible surgery coefficients and help identify the manifolds which may be surgery on knots.

Abstract: Many hidden operations in algebraic topology can be described by a theory of extended powers,related to covering spaces and the homology of the symmetric groups. Early examples are Steenrod operations in mod 2 cohomology, and Dyer-Lashof operations in mod 2 homology of classifying spaces of symmetric monoidal categories. Chas and Sullivan described operations in the rational homology of free loop spaces of manifolds; but the mod 2 homology of these spaces also have Dyer-Lashof type operations related to the homology of the braid groups. We develop a modified Q-ring theory of these operations, parallel to the Bisson-Joyal Q-ring theory for Steenrod and Dyer-Lashof operations.

Abstract: A well known result of Giroux tells us that isotopy classes of contact structures on a closed three manifold are in one to one correspondence with stabilization classes of open book decompositions of the manifold. We will introduce a characterization of tightness of a contact structure in terms of corresponding open book decompositions, and show how this can be used to resolve the question of whether tightness is preserved under Legendrian surgery.

Abstract: We're interested in the following question: fix $k>0$ and an orientable surface $S$. Given a pair of simple closed curves $(x,y)$ which fill $S$, how many curves (simple or possibly with self-intersection) intersect the union of $x$ and $y$ at most $k$ times? In joint work with S. Huang, we showed that for $k=1$, the filling pairs which maximize this quantity are the minimally intersecting ones. In this talk, we outline a result which shows that the minimally intersecting filling pairs are also close to maximizing this quantity in an asymptotic sense as k approaches infinity. The proof uses an effective version of the Masur-Minsky distance formula derived in joint work with S. Taylor and R. Webb, and also a bit of Teichmuller theory.

Abstract: I will talk about joint work with Thomas Koberda and Sam Taylor on a version for right-angled Artin groups of convex cocompactness for subgroups of mapping class groups. The latter were defined by Farb and Mosher in partial analogy to convex cocompact hyperbolic manifold groups, and there are interesting connections between all three settings.

Abstract: Aramayona and Leininger have provided a "finite rigid subset" $X(S)$ of the curve complex $C(S)$ of a surface $S$, characterized by the fact that any simplicial injection $X(S)\hookrightarrow C(S)$ is induced by a unique simplicial automorphism $C(S)\cong C(S)$. We prove that, in the case of the sphere with $n>4$ marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a $\mathrm{Mod}(S)$-module generator for the reduced homology of the curve complex $C(S)$, answering in the affirmative a question posed by Aramayona and Leininger. For the surface $S$ with genus $g>2$ and $n=0$ or $n=1$ marked points we find that the finite rigid set $X(S)$ of Aramayona and Leininger contains a proper subcomplex whose reduced homology class is a $\mathrm{Mod}(S)$-module generator for the reduced homology of $C(S)$ but which is not itself rigid. This is joint work with J. Birman and W. Menasco.