Papers

We use Khovanov-Floer theories and their spectral sequences to construct differentials on the corresponding skein-lasagna modules. Then we show that for 2-handlebodies, a spectral sequence of TQFTs coming from a Khovanov-Floer theory gives a spectral sequence of lasagna modules under the aforementioned differential. As a corollary, we recover the rank inequality between the Khovanov and Lee lasagna modules. We also prove certain properties like the connect-sum formula that these spectral sequences satisfy.

We investigate which 3-manifolds bound 4-manifolds that are homeomorphic but not stably diffeomorphic, where stabilising means taking connected sum with copies of $S^2\times S^2$. We show that every closed, orientable 3-manifold admits such fillings, as do certain families of nonorientable 3-manifolds. In contrast we show that for a 3-manifold containing a 2-sided $\mathbb{RP}^2$, any two smooth, homeomorphic fillings are stably diffeomorphic.

We classify fixed point homogeneous Riemannian orbifolds with positive sectional curvature up to equivariant diffeomorphism. As a corollary we obtain a classification of positively curved orbifolds with maximal symmetry rank.

We derive a closed formula for the Masur-Veech volumes of spin-parity components of the stratum of abelian differentials with a single zero of maximal order. Our approach is based on the intersection theory of the virtual subcone of spin-parity squares inside the Hodge bundle, recently developed by Holmes-Politopoulos-Sauvaget. This is a spin refinement of a classical result of Sauvaget and agrees with the lattice-point counting technique of Chen-Möller-Sauvaget-Zagier. We further apply this theory to compute spin counterparts of virtual volumes, defined recently by Sauvaget. These virtual volumes are related to area Siegel-Veech constants and we are able to compute these invariants component-wise. In addition, by comparing our method for non-parity squares with the classical result of Sauvaget, we compute specific Chiodo integrals.

We prove that if Zeeman's collapsibility conjecture holds for standard polyhedra, then it holds in general.

We express the local-equivalence classes of real Seiberg–Witten Floer homotopy types for spin Seifert $3$-manifolds with certain odd involutions in terms of the Fukumoto-Furuta $w$-invariant. The proof uses real orbifold Bauer-Furuta invariant for a class of spin $4$-orbifolds constructed by Fukumoto-Furuta-Ue. For torus knots, we prove that the local-equivalence classes of real Floer homotopy types associated with their $2^k$-fold cyclic branched covers are eventually periodic in $k$. Finally, we prove that the integer-valued concordance homomorphisms $\{8\delta_R^{(k)}\}_{k\geq1}$ are linearly independent.

In 1989 Olshanskii introduced a three-parameter family of coordinate-surjective homomorphisms from the genus-two surface group to a direct product of two rank-two free groups. When the common quotient of the two coordinate images is finite of order $n$, restriction to the corresponding regular cover produces an epimorphism \[ \pi_1(S_{n+1})\longrightarrow F_{n+1}\times F_{n+1}. \] We call these maps Olshanskii epimorphisms. They form an explicit high-genus test family for the standardness problem for splitting epimorphisms. We prove that every Olshanskii epimorphism is standard. The genus-two homomorphism determines a Heegaard splitting of a Seifert fibered $3$-manifold over $S^2$ with at most three exceptional fibers. In the finite-quotient cases, classical Seifert theory shows that its universal cover is $S^3$; Waldhausen's theorem then implies that the lifted genus-$(n+1)$ Heegaard splitting is standard. The proof uses neither Perelman's theorem nor the general Poincaré theorem. We also give a constructive treatment of the quaternion case $Q(2,2,2)\cong Q_8$, whose covering surface has genus nine. A maximal tree in the quaternion Schreier graph yields the covering handlebody and explicit Schreier bases. Using geometric longitude–meridian pairs and explicit surface automorphisms supported on the nine one-holed tori, we transform the lifted meridian words into a free basis. A separate fixed-rank Andrews–Curtis certificate reduces the associated balanced presentation. This paper supplies the detailed proof of the result announced in the previous Kharlampovich, Vdovina paper.

We introduce a natural class of hyperbolic surfaces called flippered surfaces that generalize crowned hyperbolic surfaces (i.e.: worldsheets for open strings). We develop their Teichmüller and moduli-space theory, construct generalized Weil-Petersson volume forms and Chekhov's action, and prove that the resulting generalized Mirzakhani volumes are finite. We establish three geometric recursion formulae-neck chopping, disk excision, and crown extraction-which express these volumes in terms of those of topologically simpler surfaces. For the fundamental polygonal and annular cases, we derive integral representations involving conical Legendre functions, as well as explicit formulae in terms of elliptic integrals and polylogarithms. We further describe the arithmetic structure of the Taylor coefficients of these volumes, show that suitable specializations are Kontsevich-Zagier periods, and prove identities at the imaginary boundary length $2\pi\sqrt{-1}$ that generalize the Do-Norbury paraphrasing of string and dilaton-type equations.

The main result of the paper shows that each connected polyhedral surface with a hyperbolic background metric, or a Euclidean background metric with uniformly bounded circumdisk radii, is discrete conformal to a complete constant-curvature Riemannian surface equipped with a nonempty closed discrete subset. We also prove a discrete Riemann mapping theorem. The proofs are based on the recent work on the discrete Schwarz lemma, the discrete Liouville theorem for polyhedral surfaces, and a Weyl-type realization theorem for hyperbolic surfaces.

We study the stars at infinity in the Thurston boundary of Teichmüller space for the Thurston and Teichmüller metrics. For the Thurston metric, we prove that for every finite-type surface, the star of a projective measured lamination is exactly its zero set, extending a theorem of Liu-Shi from closed surfaces. Moreover, the based star agrees with the star. For the Teichmüller metric, we show that both the based star and the star of a projective measured lamination coincide with its two-step zero set. This set can be strictly larger than the zero set, disproving a conjecture of Karlsson.

A multisection (as defined by Ben Aribi, Courte, Golla and Moussard) is a decomposition of a closed orientable manifold into model pieces. These model pieces are top-dimensional 1-handlebodies, whose subcollections intersect along 1-handlebodies of lower dimensions, and whose global intersection is a closed surface. This concept extends the notions of Heegaard splittings and trisections to higher dimensions. In this article, we adapt multisections to compact manifolds with boundary, generalizing sutured Heegaard splittings and relative trisections to every dimension. We define the associated diagrams, which encompass sutured Heegaard diagrams and relative trisection diagrams. A relative multisection induces a particular decomposition of the boundary of the manifold, which we call a relative fibration. We show that, in dimension n greater than 3, a connected manifold which relatively fibers is necessarily either a sphere, or a connected sum of copies of the product of the circle with the sphere of dimension n-1. We prove that a compact 5-manifold whose boundary relatively fibers admits a relative multisection. We also state a gluing theorem that allows to combine suitable relatively multisected manifolds with boundary into multisected closed manifolds.

I provide a detailed exposition of the fundamental group calculations in the “reverse engineering" construction of an exotic $\CP ^2\#3\overline{\CP }^2$ and of an exotic $\CP ^2\#2\overline{\CP }^2$.

In this note we show that the image of the third Johnson homomorphism $\tau_3$ coincides with the kernel of Morita's trace map for the case of a surface of genus $g$ with one boundary component $\Sigma_{g,1}$, when $g \geq 6$. Moreover, as a consequence, we get that the image of $\tau_3$ on the handlebody subgroups coincides with the intersection of kernels of the Morita trace map and the Lagrangian trace maps.

We generalize the notion of the Hopf link to the context of spatial graphs and handlebody-links, and study their symmetry and classification problems.

Fox's conjecture predicts that the absolute values of the coefficients of the Alexander polynomial of an alternating link form a trapezoidal sequence. Kálmán, Mészáros, and Postnikov gave a new proof of trapezoidality for special alternating links using a polynomial associated with flat vector arrangements. We prove that the flat arrangement polynomial has trapezoidal coefficients for every full-dimensional flat vector arrangement, answering the corresponding open question. Our proof gives a geometric interpretation of this polynomial in terms of convex polytopes. As an application, we prove that the Murasugi–Stoimenow polynomial of every connected Eulerian digraph has trapezoidal coefficients.

This is the third of three related preprints. We consider here which groups $\pi$ and $PD_3$-complexes $Y$ are realised by $PD_4$-pairs $(X,Y)$ with $X$ aspherical and $\pi_1X\cong\pi$, and show that such a pair may be assembled from $(D^4,S^3)$ and $PD_4$-pairs of groups, by adding 1-handles and mapping cylinders of $\mathbb{Z}\pi_1N$-homology equivalences over boundary components $N$, if and only if $H^2(\pi;\mathbb{Z}\pi)=0$. (This includes all such pairs with $\pi_1$-injective boundary components and all with $\pi$ a free group, but none with $c.d.\pi=2$. Adding a 1-handle includes connected sum.) If there is a finite 2-dimensional $K(\pi,1)$-complex then $\pi$ is realisable by some such pair $(X,Y)$, but there are no obvious building blocks analogous to $PD_4$-pairs of groups, except for when $\pi$ is a $PD_2$-group.

Let $X=G/K$ be a symmetric space of noncompact type, of dimension $n$ and rank $r$, and let $Y=\Gamma\backslash X$. Sarnak's local bound for an $L^2$-normalized spherical joint eigenfunction with regular tempered parameter of size $T$ is $\|\phi\|_\infty\ll T^{(n-r)/2}$. We prove $o(T^{(n-r)/2})$ locally uniformly on every quotient. On finite-volume real hyperbolic manifolds this is uniform in the expanding cusp range $y\leq T^\beta$, $\beta<1/2$. If the injectivity radius is bounded below, we prove the global estimate $T^{(n-r)/2}(\log T)^{-r/2}$. The proof makes use of a novel kernel argument and a uniform bound on the spherical function.

We define a $2\pi/3$-rotationally invariant 6-valent vertex in the $\mathfrak{gl}_N$ web category. When $N = 4, 5$, we provide a categorification of the 6-valent vertex using $\mathfrak{gl}_N$ foams and decompose the hexagon web into a direct sum of indecomposables. A similar decomposition is conjectured for $N \geq 6$.

We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.

We describe in detail and provide computer code for constructing optimal geometric braids of three strands from algebraic data encoding the braiding pattern. Our optimality criterion uses the known interpretation of braids as homotopy classes (rel endpoints) of paths in $SL_2(\mathbb{R})$ joining the identity $I$ to some $A \in SL_2(\mathbb{Z})$, i.e., elements of the fundamental group of $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$, a quotient equivalent to the unit tangent bundle of the classical modular surface $\mathbb{H}/PSL_2(\mathbb{Z})$. The main technical result finds the length minimizing geodesic in a prescribed homotopy class. From another perspective, of independent interest, this amounts to shooting the shortest geodesic that connects, with a prescribed number of spins en route, two given unit tangent vectors to the Poincaré (half-)plane $\mathbb{H}$. The length is measured by a Riemannian metric from a family of deformed Sasaki metrics, sometimes called Kaluza-Klein metrics, whereby unit tangent vectors can be interpreted as infinitesimal rotors, called spinners, and the ratio of the mass to the moment of inertia is the deformation parameter. At the universal covering level $\widetilde{SL}_2(\mathbb{R})$, the geometry is one of Thurston's eight model 3D geometries. In the vanishing mass limit, it converges to the better understood Carnot-Carathéodory contact geometry, where our geodesic shooting extends known formulas. The finite mass case is more delicate, requiring numerical solution of a targeting equation. The characterization of the length minimizing geodesics (No-multiplicity Theorem) and the resulting targeting equation are the main original contribution. The exposition is complete and multi-pronged, aimed at a broad spectrum of readers. (Numerous figures are the backbone of the narrative and should be viewed in color.)

We introduce 3-group-graded extensions of fusion 2-categories, where 3-groups are modeled by 2-crossed modules. From this data, we derive a state-sum invariant of 4-manifolds equipped with a homotopy class of maps to a homotopy 3-type, or equivalently of flat 3-bundles over 4-manifolds. This invariant is nontrivial and generalizes the Douglas-Reutter invariant of 4-manifolds. To construct our invariant, we encode homotopy classes of maps to the classifying space of a 3-group $\sigma$ via $\sigma$-colorings of triangulations, and we generalize Pachner's theorem in this context.

In the appendix of [LSV26], the authors produce an invariant for contact structures on $\Sigma \times [0,1]$. In this paper, we prove a result announced there which states that this is in-fact a full invariant. As an application, we provide an independent proof of Eliashbergs theorem and show that various construction methods for contact structures are equivalent.

We prove that any Busemann G-space such that every sufficiently small metric ball is convex is a topological manifold. The key ingredient in the proof is Ivanov's Helly theorem. The appendix contains a counterexample to a question of Berestovskii–Halverson–Repovš.

In this paper, we study planar virtual knotoids, which generalize virtual knots and knotoids both. We consider the problem of whether two given planar virtual knotoids can be transformed into each other via a sequence of local moves and what is the shortest sequence length. By introducing $r$-covering of planar virtual knotoids, we determine the homotopy relationship between different planar virtual knotoids, and obtain lower bounds of the Gordian distance between them. On the basis of these results, we demonstrate calculations of the exact Gordian distance of several given pairs of homotopic planar virtual knotoids. Furthermore, we investigate $\Delta$-moves and virtual region crossing changes for planar virtual knotoids, and derive lower bounds of the Gordian distance with respect to both moves via $r$-coverings.

A knot in $S^3$ is said to be not $SU(2)$-abelian knot, if every non-trivial surgery along it yields a $3$-manifold whose fundamental group admits an irreducible $SU(2)$-representation. We provide examples of knots in $S^3$ that are not $SU(2)$-abelian. A knot $K$ is said to be $SU(2)$-clean if whenever the fundamental group of the integer $r$-surgery $\fund{K(r)}$ has no $SU(2)$-irreducible representation, then the Alexander polynomial of $K$ does not vanish at any $r$-th root of unity. We show that if $K$ is a non-trivial $SU(2)$-clean knot, the connected sum $K\#K$ is never $SU(2)$-abelian. Finally, combining known results on $SU(2)$-abundant knots and a classical epimorphism between knot groups, we show that every non-torus knot with at most $9$ crossings is not $SU(2)$-abelian, with at most the two exceptions of $9_{47}$ and $9_{49}$.

We construct an explicit finite chain of non-aspherical group presentation complexes $K(\mathcal X)\subset K(\mathcal Y)\subset K(\mathcal Z)$ in which $K(\mathcal Y)$ is Cockcroft, both inclusion-induced maps on $\pi_2$ are zero, and the pair $(K(\mathcal Y),K(\mathcal X))$ has the identity property. This answers a question of Hitchman. The construction is given directly from a short balanced presentation of the binary icosahedral group. As the second result, we construct an integral Lie ring presentation in which a subpresentation has a nontrivial identity among relations, whereas the presentation itself has none.

A unit-area translation surface is called maximal if it maximizes the length of the shortest saddle connection among all surfaces in the same stratum. We investigate whether a non-maximal translation surface can be continuously deformed into a maximal one while the systole increases strictly monotonically. Although such a deformation does not exist in general due to the existence of local but not global maxima of the systole function, we prove that it exists for square-tiled surfaces and translation surfaces obtained from regular hexagons and regular octagons. In each case, we explicitly construct a continuous deformation to a maximal translation surface along which the systole is strictly increasing. Finally, the preceding construction yields another family of translation surfaces admitting continuous systole-increasing deformations to maximal surfaces.

This article introduces and studies the entropy spectrum of a hyperbolic surface, that is the set of entropies of its subsurfaces. The main results are that the entropy spectrum is a reverse well-ordered multiset, with finite multiplicities, and that there is a quantifiable gap around the value $1$. This gap comes from a counting result on the number of non-filling geodesics which in turn comes from explicit estimates on the number of curves on surfaces with boundary in terms of geometric data. The geometric data includes lengths of boundary geodesics and so-called boundary width, which measures maximal distance to the boundary but can also be interpreted in terms of the topology of the surface and the systole.

We study the behavior of good involutions under two fundamental constructions of quandles: interaction-free unions and direct products. In particular, we show that the set of good involutions on a quandle decomposes naturally into the set of involutions on its maximal trivial component and the set of good involutions on the complement. Moreover, we construct an example of a connected noninvolutory symmetric quandle with multiple good involutions, which serves as a counterexample to a conjecture by Ta.

Lee and Marquis exhibited convex cocompact hyperbolic reflection groups in dimension 5 whose limit sets are homeomorphic to the 3-sphere, but none of whose finite-index subgroups can be realized as 4-dimensional real hyperbolic lattices. Using different methods, we furnish right-angled examples.