In joint work with Xujia Chen and Sander Kupers, we construct a bracket operation on the space of framed disk bundles of fiber dimension at least 4. Kontsevich used integrals over configuration spaces to produce graph homology classes from classes of disk bundles. We prove that our bracket operation is compatible with these Kontsevich characteristic classes via the bracket operation on graph homology. Applying our bracket to Watanabe’s bundles from Borromean surgery on trivalent graphs, we obtain new disk bundles, some of which are nontrivial.
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We begin by introducing the Torelli subgroup of the mapping class group of a surface and outlining known results about its low-dimensional cohomology. We then present recent work extending these results to a Torelli subgroup of the mapping class group of a handlebody.
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The Borel conjecture predicts that closed, aspherical manifolds (i.e., those with contractible universal cover) are topologically rigid: they are determined up to homeomorphism by their fundamental group. I will discuss the smooth version of this conjecture (concerning manifolds up to diffeomorphism), which is true in dimensions ≤ 3 but long known to be false in all dimensions ≥ 5. I will explain joint work with Davis, Huang, Ruberman, and Sunukjian that resolves the remaining 4-dimensional case by detecting exotic smooth structures on certain closed aspherical 4-manifolds.
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Gromov defined macroscopic dimension of metric spaces to study manifolds admitting a Positive Scaler Curvature (PSC) metric, via their largeness properties. He conjectured universal cover of PSC n-manifolds should have macroscopic dimension at most n-2. This conjecture depends heavily on the fundamental group of the n-manifold. Under the assumption of the Strong Novikov Conjecture, we prove that closed spin manifolds having fundamental group a Graph Product of geometrically finite groups satisfies the conjecture, provided the vertex groups have classifying space which becomes wedge sum of Moore Spaces, after finitely many suspension. This generalizes our previous result when the fundamental group is a RAAG. We developed a crucial property called 1-Step Stabilization Property (1-SSP) for groups to prove the above. We also found an examples of groups not satisfying 1-SSP, inspiring more investigation towards possible counter example related to Gromov’s conjecture.
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The most basic Riemannian manifolds are those admitting a complete metric of constant curvature. The classification of closed manifolds with metrics of positive and zero curvature is has been relatively well-understood for quite a long time. Constant curvature -1 manifolds, hyperbolic manifolds, remain quite a bit more mysterious, particularly in high dimensions. I will give a (biased) narrative regarding what we know, including a number of exciting recent results with connections to dynamics and geometric group theory, and look forward to some problems I hope to see solved in the coming years.
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The study of surface subgroups in 3-manifolds has drawn sustained attention for decades, motivated both by their intrinsic geometric richness and by their broad consequences in geometric topology, geometric group theory, and dynamics. A landmark result is the Surface Subgroup Theorem of Kahn–Markovic, which states that every cocompact Kleinian group contains a ubiquitous collection of closed surface subgroups. In this talk, we will introduce some key developments in the subject and highlight our recent progress, including joint work with Jeremy Kahn, and with Xiaolong Han and Jia Wan.
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Works of Donaldson and Gompf show that a closed, oriented 4-manifold admits a symplectic structure if and only if it admits the structure of a Lefschetz pencil. However, the question of how many Lefschetz pencils (or fibrations) a given symplectic 4-manifold admits remains open. Works of Park--Yun and Baykur construct 4-manifolds admitting arbitrarily large (but finite) numbers of Lefschetz pencils or fibrations of the same genus. In this talk, we will construct infinitely many inequivalent Lefschetz pencils of the same genus on ruled surfaces of negative Euler characteristic. In fact, our construction gives the first example of infinitely many inequivalent but diffeomorphic Lefschetz pencils and fibrations of the same genus. This is joint work with Carlos A. Serván.
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Given a finite group action on a manifold, we discuss the following question: if two equivariant diffeomorphisms are isotopic, must they be equivariantly isotopic? In the case of closed hyperbolic surfaces, a remarkable theorem of Birman and Hilden says that the answer is “yes”: isotopy implies equivariant isotopy. By contrast, we show that in dimensions three and higher, there are many diffeomorphisms which are isotopic but not equivariantly isotopic. We will explain the new obstructions that arise in higher dimensions, as well as some applications and further questions that don’t arise in the world of surfaces.
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Let $\textbf{MCG}(X)$ denote the group of isotopy classes of self-homeomorphisms of a space $X$. When $S$ is an orientable infinite type surface $S$ with no planar ends and without boundary, the extended mapping class group $\textbf{MCG}(S)$ is isomorphic to $\textbf{MCG}\left(\overline{S}\right)$ where $\overline{S}$ is the Freudenthal compactification of $S$. Using this identification, it follows that $\textbf{MCG}(S)$ canonically embeds into $\text{Out}\left(\pi_1\left(\overline{S}\right)\right)$. It remains open if $\textbf{MCG}(S)$ is isomorphic to $\text{Out}\left(\pi_1\left(\overline{S}\right)\right)$ in the spirit of the Dehn-Neilsen-Baer Theorem. A clear difficulty is the fact that $\overline{S}$ is not locally simply connected and $\pi_1\left(\overline{S}\right)$ is uncountable and not free. In this talk, we will show that $\overline{S}$ is the quotient of $\mathbb{D}^2$ by a countable edge-pairing on $\mathbb{S}^1$. This structural decomposition implies that $\overline{S}$ may constructed by attaching a single 2-cell to a one-dimensional Peano continuum. Using established technology for dealing with fundamental groups of one-dimensional Peano continua, we show that $\pi_1\left(\overline{S}\right)$ is the free product with amalgamation of two locally free groups along an infinite cyclic group. We also show that every automorphism $\phi:\pi_1\left(\overline{S}\right)\to\pi_1\left(\overline{S}\right)$ is induced by a continuous map $f:\overline{S}\to \overline{S}$ that restricts to a homeomorphism on the end set $\overline{S}\backslash S$.
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We compute the image of the monodromy representation (as a subgroup of the mapping class group) associated to a complete linear system of curves in a simply connected smooth projective surface X under some ampleness hypotheses. It turns out to always be of finite index.
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In 1997, Sarnak conjectured that the determinant of the Laplacian is a Morse function on the space of unit area Riemannian metrics on a given real surface, and hence induces a Morse function on its moduli space. Meanwhile, the systole function, defined as the length of a shortest essential closed geodesic with respect to the base Riemannian metric, is topologically Morse on the Teichmüller space of n-dimensional flat tori (due to Ash) and of Riemann surfaces of genus g with n marked points (due to Akrout), though it does not yield a classical Morse theory. In this talk, I will introduce a family of Morse functions, denoted sys_T, defined as weighted exponential averages of all geodesic-length functions, on the Deligne--Mumford compactification (M_{g,n} bar). These functions are compatible with the Deligne--Mumford stratification and the Weil--Petersson metric, and their critical points can be characterized by a combinatorial property named eutaxy. I will talk about the index gap theorem for sys_T and its homological consequences, in the form of a stability theorem for the homology of moduli spaces of stable curves. I will also briefly explain how sys_T connects to Sarnak’s conjecture.
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Gromov and Thurston famously used hyperbolic branched cover manifolds to construct the first examples of manifolds which admit a pinched negatively curved metric, but do not admit any locally symmetric metric. Much more recently, Fine and Premoselli (n=4) and Hamenstadt and Jackel (n>4) proved that many of these hyperbolic branched covers admit negatively curved Einstein metrics. In this talk I will give an overview of these results and show how, in joint work with Lafont, we extended the construction of Fine and Premoselli to complex hyperbolic branched covers. This gives an explicit description of the first known negatively curved Kahler-Einstein metric on a manifold which does not admit a locally symmetric metric, whose existence was first proved by Guenancia and Hamenstadt.
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The Zimmer program seeks to classify smooth actions of arithmetic groups, like SL(n,Z), on compact manifolds. Separately, the Nielsen realization problem asks when a subgroup of a mapping class group Mod(M) can be realized by a group of diffeomorphisms of M. In many natural situations, the mapping class group is closely related to an arithmetic group, and the realization problem is tied to the Zimmer program. I will discuss examples of this connection and describe some recent results and open questions.
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Pseudo-isotopy is an equivalence relation on homeomorphisms that lies between isotopy and homotopy. Classifying homeomorphisms of 4-manifolds up to pseudo-isotopy is a potentially tractable problem, whereas isotopy classifications currently elude us outside of the simply-connected case. I will explain a program to understand some of this difference using the smooth invariants of Hatcher-Wagoner and Igusa. A result is the construction of many examples of homeomorphisms that are pseudo-isotopic to the identity but not isotopic to the identity on a range of 4-manifolds, including the 4-torus. This is joint work with Isacco Nonino.
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Given a reduced plumbing tree and a spin-c structure, I will discuss how to construct a plumbed 3-manifold invariant in the form of a Laurent series twisted by a root lattice. Such a series is invariant under the Neumann moves on plumbing trees and the action of the Weyl group. These series-valued invariants generalize the Z-hat series of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park and Ri. They are motivated by the study of the WRT invariants, and the work of Akhmechet-Johnson-Krushkal which found connections with lattice cohomology. Time permitting, I will also discuss a multivariable generalization of the root lattice-twisted series for knot complements and gluing formulas. This is joint work with N. Tarasca.
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3-dimensional pseudomanifolds are CW-complexes with the property that the link of each point is a closed, orientable surface. We give some interesting examples of these, show that there are many examples of these groups virtually algebraically fibering, and give some applications to higher-dimensional Coxeter groups. This is joint work with Lorenzo Ruffoni.
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The existence and nonexistence of tight contact structures on the 3-manifold are interesting and important topics studied over the past thirty years. Etnyre-Honda found the first example of a 3-manifold that does not admit tight contact structure, and later Lisca-Stipsicz extended their result and showed that a Seifert fiber space admits a tight contact structure if and only if it is not the smooth (2n − 1)-surgery along the T(2,2n+1) torus knot for any positive integer n. Surprisingly, since then no other example of a 3-manifold without tight contact structure has been found. Hence, it is interesting to study if all such manifolds, except those mentioned above, admit a tight contact structure. Towards this goal, I will discuss the joint work with Zhenkun Li and Hugo Zhou about showing any negative surgeries on any knot in S^3 admit a tight contact structure.
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It is known that every oriented plane field on a closed 3-manifold is homotopic to an integrable one. However, this no longer holds if one requires the foliation to be taut. This leads naturally to the question of which second cohomology classes can arise as the Euler classes of co-oriented taut foliations on a given 3-manifold M. When M is a rational homology sphere, the second cohomology group is finite, and the zero class plays a distinguished role. In this talk, we present infinitely many rational homology 3-spheres, including small Seifert fibred, hyperbolic, and toroidal examples, that admit co-oriented taut foliations but do not admit any with vanishing Euler class. We will also discuss the implications of these examples in the context of the L-space conjecture. This is joint work with Steve Boyer, Cameron Gordon and Duncan McCoy.
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Kielak and Fisher have connected the $L^2$-Betti numbers (and their finite field variants) to the Novikov homology for a RFRS group $G$. This in turn relates vanishing of $F$-$L^2$-Betti numbers of $G$ to algebraic virtual fibering of $G$. We will give an example of a RFRS group $G$ which has vanishing top-dimensional Novikov cohomology with all field coefficients but not with $\mathbb{Z}$-coefficients.
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In a joint work with Daren Chen and Ian Zemke, we study the torsion order of Heegaard Floer homology under L-space satellite operators, which by a result by Alishahi-Eftekhary, leads to an unknotting number bound. The argument resembles the work of Hom-Lidman-Park; instead of using immersed curves, we use the L-space satellite formula by Chen-Zemke-Zhou.
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