几何拓扑

Let $M$ be a compact 3--manifold with boundary a single torus. We present upper and lower complexity bounds for closed 3--manifolds obtained as even Dehn fillings of $M.$ As an application, we characterise some infinite families of even…

Isotopy classes of diffeomorphisms of the 4-sphere can be described either from a Cerf theoretic perspective in terms of loops of 5-dimensional handle attaching data, starting and ending with handles in cancelling position, or via certain…

We prove that the knots and links that admit a 3-highly twisted irreducible diagram with more than two twist regions are hyperbolic. This should be compared with a result of Futer-Purcell for 6-highly twisted diagrams. While their proof…

We show that ribbon rational homology cobordism is a partial order within the class of irreducible 3-manifolds. This makes essential use of the methods recently employed by Ian Agol to show that ribbon knot concordance is a partial order.

We associate to every positive braid a braid monodromy group, generalizing the geometric monodromy group of an isolated plane curve singularity. If the closure of the braid is a knot, we identify the corresponding group with a framed…

According to Bestvina-Bromberg-Fujiwara, a finitely generated group is said to have property (QT) if it acts isometrically on a finite product of quasi-trees so that orbital maps are quasi-isometric embeddings. We prove that the fundamental…

Using ideas from 3-manifolds, Hatcher--Wahl defined a notion of automorphism groups of free groups with boundary. We study their Torelli subgroups, adapting ideas introduced by Putman for surface mapping class groups. Our main results show…

We consider a class of right-angled Coxeter orbifolds, named as simple orbifolds, which are a generalization of simple polytopes. Similarly to manifolds over simple polytopes, the topology and geometry of manifolds over simple orbifolds are…

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric triangulation into hyper-ideal hyperbolic tetrahedra. So far, this conjecture had only been proven for a few special 3-manifolds. In this…

On a connected surface $N$ with negative Euler characteristic, the free homotopy class of a loop obtained by smoothing an intersection of two closed geodesics may wind around a puncture. Chas and Kabiraj showed that this phenomenon does not…

We describe five ideal triangulations of the 3-cusped hyperbolic `magic manifold' that are each compatible with well-established techniques for triangulating Dehn fillings. Using these techniques, we construct low-complexity triangulations…

Given a branched cover of manifolds, one can lift homeomorphisms along the cover to obtain a (virtual) homomorphism between mapping class groups. Following a question of Margalit-Winarski, we study the injectivity of this lifting map in the…

Motivated by an observation of Dehornoy, we study the roots of Alexander polynomials of knots and links that are closures of positive 3-strand braids. We give experimental data on random such braids and find that the roots exhibit marked…

We present an algorithm that computes Friedl and L\"uck's twisted $L^2$-Euler characteristic for a suitable regular CW complex, employing Oki's matrix expansion algorithm to indirectly evaluate the Dieudonn\'e determinant. The algorithm…

Recent constructions have shown that interesting behaviours can be observed in the finiteness properties of K\"ahler groups and their subgroups. In this work, we push this further and exhibit, for each integer $k$, new hyperbolic groups…

We construct a generalization of the Seiberg-Witten Floer spectrum for suitable three-manifolds $Y$ with $b_1(Y)>0$. For a cobordism between three-manifolds we define Bauer-Furuta maps on these new spectra, and additionally compute some…

In this paper, we develop new techniques for understanding surfaces in $\mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply…

We describe an algorithm that, given a 3-manifold M, outputs a finite set containing all minimal volume k-component hyperbolic link complements in M. A key step, that might be of independent interest, is an algorithm that, given two…

We introduce framed versions of the $L$-moves and prove a one move theorem for the extension of the Markov theorem for framed braids. We further introduce framed versions of the Hilden and Pure Hilden groups, we give presentations and we…

Heegaard splittings stratify 3-manifolds by complexity; only $S^3$ admits a genus-zero splitting, and only $S^3$, $S^1 \times S^2$, and lens spaces $L(p,q)$ admit genus-one splittings. In dimension four, the second author and Jeffrey Meier…