几何拓扑

Stone's representation theorem asserts a duality between Boolean algebras on the one hand and Stone space, which are compact, Hausdorff, and totally disconnected, on the other. This duality implies a natural isomorphism between the…

We consider random walks on semisimple Lie groups where the support of the step distribution generates (as a group) a Zariski dense discrete subgroup of infinite covolume. When the semisimple Lie group has property (T), we show that the…

We introduce a diagrammatic approach to Rasmussen's $s$-invariant, based on Bar-Natan's reformulation of Khovanov homology for tangles and cobordisms. This method enables a local computation of $s$ from a tangle decomposition of a knot…

Let $\Sigma$ be a closed orientable surface of genus at least two, and let $X, Y$ be distinct marked Riemann surface structures on $\Sigma$, possibly with opposite orientations. In this paper, we show that there are (exactly) countably…

We present new smoothing techniques for topologically embedded surfaces in smooth 4-manifolds, which give topological isotopy to a smooth surface. As applications, we prove "topological = smooth" results in dimension 4 for certain disks and…

We use annular foam TQFTs introduced by the first two authors to define equivariant $SL(2)$ and $SL(3)$ web algebras in the annulus. To a diagram of a tangle in the thickened annulus we assign a complex of bimodules over these algebras…

We show that the largest size of a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $\chi$ grows quadratically in $|\chi|$. This resolves a longstanding question of…

When $\pi:\widetilde{\Sigma}\rightarrow D^2$ is a cover of the disc branched over $n$ marked points, the braid group $B_n$ acts on the disc by homeomorphisms fixing the marked points setwise. A braid $\beta$ \textit{lifts} if there is a…

Given a maximal geodesic lamination $\lambda$ on a closed oriented surface $S$ of genus $g$, the space of $d$-pleated surfaces with pleating locus $\lambda$ is an open subset of $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ obtained…

Let $\Sigma$ be a compact, orientable surface of genus $g$, and let $\Gamma$ be a relation on $\pi_0(\partial \Sigma)$ such that the prescribed arc graph $\mathcal{A}(\Sigma,\Gamma)$ is Gromov-hyperbolic and non-trivial. We show that…

We propose two definitions of configuration Lie groupoids and in both the cases we prove a Fadell-Neuwirth type fibration theorem for a class of Lie groupoids. We show that this is the best possible extension, in the sense that, for the…

Two people who pioneered the study of mapping class group-equivariant deformation retractions of Teichm\"uller space of closed compact surfaces are Schmutz Schaller and Thurston. This paper studies how the two different approaches are dual…

We prove that the cut-system complex of a sutured compression body, with vertices representing cut-systems and edges corresponding to handleslides, becomes simply connected when six kinds of 2-cells are attached. Moreover, we define tight…

We prove the existence of a localization spectral sequence for the hat variant of Guth and Manolescu's recent construction of real Heegaard Floer homology, and apply it to branched double covers and strongly invertible knots. Our…

We determine a simple condition on a particular state graph of an alternating knot or link diagram that characterizes when the unoriented genus and crosscap number coincide, extending work of Adams and Kindred. Building on this same work…

The first main results of this note establish forms of the hyperbolic laws of cosines and sines for certain classes of quadrilaterals and pentagons in the hyperbolic plane, having at least one ideal vertex and right angles at non-ideal…

In our earlier work on $2$-torsion in instanton Floer homology, we considered only integral surgeries on a knot $K\subset S^3$ and showed that the absence of $2$-torsion forces $K$ to be fibered. The present paper extends the result to all…

We define a new algebraic structure called a \emph{pointed rack} and use it to construct ambient isotopy invariants of $ n $-braids. We first introduce an integer-valued invariant of braids using pointed racks. This is then strengthened by…

We show that the open genus 2 handlebody admits uncountably-many fibrations over the circle with fiber homeomorphic to the Cantor tree surface with non-conjugate monodromies in the mapping class group. The construction generalizes to…

Let $M$ be a closed, 3-connected, 8-dimensional smooth manifold. In this paper, we compute the concordance inertia group of the product manifold $M\times\mathbb{S}^k$ for $1\leq k\leq 14$ and classify all smooth manifolds homeomorphic to…