Today's listing: 3 new submissions and 7 replacements (resubmissions). Every paper is covered below in full — original title, abstract, and structured commentary.

Authors: Valentina Bais, Riccardo Piergallini, Daniele Zuddas

Abstract (original): We show that, given and two closed connected oriented PL $4$-manifolds $M$ and $N$ such that $N$ has a handle decomposition with no $1$- and $3$-handles, there exists a $d$-fold simple branched covering if and only if there is an isometric embedding of intersection lattices . Moreover, if such $p$ exists, one can build it in such a way that its branch set is locally flat PL embedded if and has at most nodal singularities if $d = 4$.

Key result. The paper gives a complete algebraic characterisation of when one closed oriented PL 4-manifold covers another (of a specific topological type) as a simple branched cover of degree : the necessary and sufficient condition is the existence of an isometric embedding of intersection lattices . It also controls the singularity type of the branch set, showing it can be taken locally flat for and nodal for $d = 4$.

Techniques and tools. The argument is squarely in the PL topology of 4-manifolds. The central tool is intersection form theory: the intersection lattice of a simply connected 4-manifold is a complete invariant (by Freedman, in the topological category), and the authors translate the branching condition into a lattice-embedding problem. The hypothesis that $N$ has a handle decomposition without 1- or 3-handles (i.e., $N$ is "built from 0- and 2-handles only," making it resemble a -type building block) is used to control the local monodromy. Simple coverings (the monodromy group acts by transpositions) are easier to construct and the simplicity assumption feeds directly into stabilisation arguments that let the authors upgrade the branch set's regularity.

Proof idea. The strategy is a branched-cover version of the classical stabilisation method. Starting from an algebraic embedding , one constructs an initial branched covering (possibly with complicated branch locus) and then performs local modifications — pushing branch-locus self-intersections through controlled moves — to achieve the stated regularity for . The $d = 4$ vs. dichotomy in the branch-set conclusion mirrors a known threshold in branched cover theory where extra degree gives extra room to isotope away genuine singularities.

Authors: Ken'Ichi Ohshika, Athanase Papadopoulos

Abstract (original): We prove several new results on the combinatorial structures of the unit spheres of the norms induced by Thurston's metric on the tangent and cotangent spaces of the Teichmüller space of a closed surface of negative Euler characteristic. These results include a formula for the dimension of every face of a unit sphere in the tangent space in terms of an invariant of the chain-recurrent lamination representing the face. We then prove that the combinatorial structure of such a unit sphere is independent of the underlying point in Teichmüller space. Provided the genus of the surface is , we show that there is a natural isomorphism between the extended mapping class group of the surface and the group of combinatorial automorphisms of such a unit sphere. In the case of genus 2, we obtain a natural epimorphism between the two groups whose kernel is the class of the hyperelliptic involution. Regarding the unit spheres of Thurston's metric in the cotangent spaces, we obtain a formula describing the codimensions of faces of such a sphere in terms of corresponding projective measured laminations. We then give a necessary and sufficient condition for a face to be exposed, and for a face to correspond to a projectively weighted multi-curve. Some of the results obtained answer open questions.

Key result. The paper resolves the combinatorial geometry of the unit spheres in both the tangent and cotangent spaces of Teichmüller space under Thurston's (asymmetric) Finsler metric. The headline results are: (1) a dimension formula for each face of the tangent unit sphere, expressed via an invariant of the associated chain-recurrent lamination; (2) a rigidity theorem — the combinatorial type of the unit sphere does not depend on the base point in Teichmüller space; and (3) an identification of the combinatorial automorphism group of the unit sphere with the extended mapping class group (with a genus-2 exception where the kernel is the hyperelliptic involution).

Techniques and tools. The paper sits at the interface of Teichmüller theory, train-track combinatorics, and convex geometry. Thurston's metric is defined via Lipschitz constants of surface maps; its unit balls are polyhedral objects whose faces correspond to geodesic laminations and multicurves. The key combinatorial invariant is the chain-recurrent lamination (the recurrence class of the lamination measured by the metric direction). Automorphism rigidity for the unit sphere is proved by connecting combinatorial symmetries to the action of the extended mapping class group — a technique analogous to Ivanov's theorem for the curve complex.

Proof idea. For the tangent-space formula, the authors identify each face of the unit sphere with a stratum in the space of measured laminations and compute its dimension using combinatorial data of the chain-recurrent lamination (number of complementary regions and topological type). For the automorphism theorem (genus ), they show that any combinatorial automorphism must preserve the face structure and hence descend to a homeomorphism of the underlying surface, and then invoke the Dehn–Nielsen–Baer theorem to identify such homeomorphisms with mapping classes. The cotangent-sphere analysis proceeds dually, using projective measured laminations and an exposed-face criterion from convex analysis.

Authors: Marc Kegel, Eric Stenhede, Vera Vértesi

Abstract (original): A theorem of Ding and Geiges states that every closed, connected contact 3-manifold can be obtained from the standard tight contact 3-sphere by contact -surgery along a Legendrian link. The literature also contains some examples of contact Kirby moves, i.e. explicit operations on front projections of Legendrian surgery links that change the surgery link but preserve the contactomorphism type of the surgered manifold. Among the most commonly used are cancelling pairs and contact handle slides; however, these moves alone are not sufficient to relate all contact surgery diagrams of contactomorphic contact manifolds.

In this article, we introduce two new families of contact Kirby moves, called lantern moves and chain moves, and use them to give a complete set of contact Kirby moves. More precisely, we show that two contact surgery diagrams represent contactomorphic contact manifolds if and only if they are related by a sequence of planar isotopies, Legendrian Reidemeister moves, insertions or removals of standard cancelling pairs, the two standard contact handle slides, the standard lantern move, and the standard chain move. All these moves are explicit diagrammatic operations in the front projection.

The proof follows an approach initiated by Avdek through his ribbon-move framework, which is rooted in the Giroux correspondence, and combines it with a presentation by Gervais of the mapping class group. We also discuss several consequences of the main theorem, illustrating the effectiveness of the contact Kirby calculus by recovering the invariance of Gompf's -invariant purely diagrammatically and as a corollary, by deriving the topological Kirby theorem from contact-geometric methods.

Key result. This paper completes the contact Kirby calculus: it provides a finite, explicit list of moves on front-projection diagrams of Legendrian surgery links such that two diagrams represent contactomorphic manifolds if and only if they are related by a sequence from this list. The two new moves — the lantern move and the chain move — are precisely what was missing; cancelling pairs and contact handle slides alone do not suffice.

Techniques and tools. The three key ingredients are: (1) the Giroux correspondence (open book decompositions ↔ contact structures), which converts contact surgery to a problem about mapping class groups of surfaces; (2) Avdek's ribbon-move framework, which translates Giroux's correspondence into diagrammatic moves on Legendrian links; and (3) the Gervais presentation of the mapping class group, which gives a finite generating set (Dehn twists along a specific collection of curves) that matches the diagrammatic moves term-by-term. The completeness of the contact Kirby moves then follows from the completeness of the Gervais presentation.

Proof idea. Given two surgery diagrams for contactomorphic manifolds, the Giroux correspondence produces two open books with isotopic contact structures; a theorem of Giroux says these are stably equivalent. Stable equivalence translates, via Avdek's framework, into the ribbon moves. Gervais's presentation shows that all mapping class group relations lift to the diagrammatic moves (the lantern and chain relations correspond exactly to the two new moves). The "if" direction is verified by checking, case-by-case, that each listed move preserves the contactomorphism type. The paper also notes that the topological Kirby theorem can be recovered as a quotient — collapsing the contact data — which is a pleasing conceptual bonus.

Authors: Valentina Bais, Riccardo Piergallini, Daniele Zuddas

Abstract (original): We show that every closed connected non-orientable PL 4-manifold $X$ is a simple branched covering of . We also show that $X$ is a simple branched covering of the twisted -bundle if and only if the first Stiefel–Whitney class admits an integral lift. In both cases, the degree of the covering can be any number , provided that $d$ has the same parity as the Stiefel–Whitney number in the case of . Moreover, the branch set can be assumed to be non-singular if and to have just nodal singularities if $d = 4$.

Key result. Every closed connected non-orientable PL 4-manifold is a simple branched covering of (with a parity condition on the degree) or of (when admits an integral lift). This is the non-orientable counterpart of classical results by the same authors on orientable 4-manifolds (including today's new submission arXiv:2605.26337), completing the picture for all closed PL 4-manifolds.

Techniques and tools. The non-orientable setting requires characteristic class data that the orientable case does not. The Stiefel–Whitney class (and whether it lifts to an integer cohomology class) determines which target manifold is available; the parity of determines which degrees work over . The actual covering constructions again use handle decompositions, PL local models for branching, and stabilisation to improve the branch set.

Proof idea. Analogous to the orientable case but with additional characteristic class bookkeeping. The parity constraint on the degree over arises from matching the obstruction classes of the covering map; the integral-lift condition for the case ensures that the monodromy representation is compatible with the orientable double cover structure of $X$. Branch-set regularity is then improved degree-by-degree via the same stabilisation arguments.

Authors: Yonghan Xiao

Abstract (original): We show that for a real rational homology sphere $Y$ equipped with a real structure , the real monopole Floer homology defined by Li and the real Seiberg–Witten Floer homology defined by Konno, Miyazawa and Taniguchi are isomorphic. As corollaries, we identify some Froysho̊v-type invariants and prove two Smith-type inequalities.

Key result. The two competing constructions of "real" Seiberg–Witten Floer homology — Li's version and the Konno–Miyazawa–Taniguchi version — are isomorphic as invariants of real rational homology spheres with real spin^c structures. As immediate corollaries, the Frøyshov-type invariants extracted by either construction agree, and two Smith-type inequalities (relating the homology of a fixed-point set to that of the ambient manifold under the real involution) follow.

Techniques and tools. Real (or equivariant) Seiberg–Witten theory starts with a 3-manifold $Y$ equipped with an orientation-reversing involution; "real structures" on spin^c data must be compatible with the involution. Two separate Floer packages have been built on this foundation (Li's using a direct monopole-equation approach; Konno–Miyazawa–Taniguchi using Pin(2)-equivariant methods). The proof of isomorphism likely passes through a chain-level comparison of the Floer complexes, relating the analytical perturbation schemes used by each construction.

Proof idea. The isomorphism is established by constructing a chain map between the two Floer complexes and checking it is a quasi-isomorphism. The key step is identifying the generators (critical points of the Chern–Simons–Dirac functional under the real constraint) on both sides and matching the differential counts (moduli spaces of real monopoles). Corollary consequences — Frøyshov invariants and Smith inequalities — then follow from the structural properties of the shared Floer package.

Authors: Masato Tanabe

Abstract (original): Thom polynomials are universal cohomological obstructions to the appearance of singularities of given types in differentiable maps. As an application, various invariants of immersions have been expressed in terms of singularities of their extension maps, known as singular Seifert surfaces. To place these results in a unified framework, we aim in this paper to establish a relative version of Thom polynomial theory. Our results consist of four parts. (1) We introduce Thom polynomials relative to prescribed maps around the boundary (or a closed codimension-zero submanifold) that avoid singularities of given types. (2) We show a structure theorem for Thom polynomials relative to framable immersions. It expresses them as the sum of the term obtained by substituting Kervaire's relative characteristic classes into the absolute Thom polynomial and a universal correction term. (3) We determine correction terms in several cases, not only reinterpreting earlier works as instances of relative Thom polynomials but also recovering some of them. Most earlier formulas are summarized as the vanishing of correction terms. (4) We give suggestive evidence for the relative Thom polynomials of multi-singularity types , with an application.

Key result. A relative version of Thom polynomial theory is constructed, generalising the classical universal cohomological obstruction for singularities from maps on closed manifolds to maps on manifolds-with-boundary, where the boundary data (the "prescribed map") is fixed. The structure theorem expresses relative Thom polynomials as the absolute Thom polynomial with Kervaire's relative characteristic classes substituted in, plus a correction term; in many geometric cases the correction term vanishes and earlier ad hoc results become unified instances.

Techniques and tools. Classifying spaces and universal singularity theory (Thom–Mather, Boardman) provide the setting. Relative characteristic classes (Kervaire's construction) handle the boundary constraint. The paper uses spectral sequence arguments and characteristic class computations to pin down the correction terms; the multi-singularity evidence likely involves explicit enumerative geometry or intersection theory on jet bundles.

Proof idea. The relative Thom polynomial is defined via the classifying map of the relative jet bundle (jet bundle with boundary condition). The structure theorem follows by splitting the relative jet space into an absolute part and a boundary-correction part; naturality of characteristic classes forces the splitting to take the stated form. Correction terms are then computed case-by-case by identifying them as obstructions measured by Kervaire classes in specific singularity strata.

Authors: Cindy Zhang

Abstract (original): A fundamental result in 4-manifold topology asserts that any two exotic smooth structures on a simply-connected, closed 4-manifold differ by a cork twist: the operation of removing a compact, contractible, codimension-zero submanifold and regluing it by a diffeomorphism of its boundary. In this paper, we introduce the notion of a surface cork, an analogous object in the setting of smoothly embedded, closed surfaces $F$ in closed 4-manifolds $X$. This is a compact, contractible, codimension-zero submanifold intersecting $F$ in a controllable manner, whose removal and regluing via a diffeomorphism of its boundary changes the diffeomorphism type of $(X, F)$ as a pair while leaving its homeomorphism type unchanged. The way in which the surface $F$ interacts with the codimension-zero submanifold leads us to define three distinct notions of surface corks: enclosing surface corks, exterior surface corks, and transverse surface corks. We establish the existence of exterior surface corks for certain previously known examples of exotic pairs. Furthermore, we give the first explicit construction of a transverse surface cork for certain exotic families arising from Fintushel–Stern rim surgery. Notably, this transverse surface cork turns out to be diffeomorphic to a 4-ball.

Key result. This paper introduces and develops the concept of a surface cork — the analogue of the classical cork for exotic smooth structures on 4-manifolds, but now in the relative setting of exotic smoothly embedded surfaces. Three flavours (enclosing, exterior, transverse) are defined according to how the cork meets the surface $F$. The headline construction is the first explicit transverse surface cork for exotic surface families from Fintushel–Stern rim surgery; it is diffeomorphic to a 4-ball, making it as simple as possible.

Techniques and tools. The foundational input is cork theory for 4-manifolds (Akbulut–Yasui, etc.). Fintushel–Stern rim surgery is the main source of examples: it modifies a surface inside a 4-manifold by performing surgery along a torus in a neighbourhood of the surface, producing exotic pairs and that are homeomorphic but not diffeomorphic as pairs. Kirby calculus, handle diagrams, and Whitney disc arguments are used to identify and manipulate the cork submanifold.

Proof idea. For the transverse surface cork, the author identifies a 4-ball that meets $F$ transversely in a prescribed pattern, and shows that regluing by a specific diffeomorphism (one that acts nontrivially on the linking data of ) exchanges and while leaving $X$ fixed. The diffeomorphism of used is an element of the smooth mapping class group of that acts nontrivially on the ribbon/link data encoding the surface intersection, but is isotopic to the identity topologically — which is what drives the exotic phenomenon.

Authors: Mikhail Lyubich, Sabyasachi Mukherjee

Abstract (original): The goal of this survey is to present intimate interactions between four branches of conformal dynamics: iterations of anti-rational maps, actions of Kleinian reflection groups, dynamics generated by Schwarz reflections in quadrature domains, and algebraic correspondences. We start with several examples of Schwarz reflections as well as algebraic correspondences obtained by matings between anti-rational maps and reflection groups, and examples of Julia set realizations for limit sets of reflection groups (including classical Apollonian-like gaskets). We follow up these examples with dynamical relations between explicit Schwarz reflection parameter spaces and parameter spaces of anti-rational maps and of reflection groups. These are complemented by a number of general results and illustrations of important technical tools, such as David surgery and straightening techniques. We also collect several analytic applications of the above theory.

Key result. This is a survey paper rather than a research paper with a single new theorem. Its contribution is a unified conceptual framework connecting four separate sub-fields of conformal dynamics — anti-rational map iteration, Kleinian reflection groups, Schwarz reflection dynamics in quadrature domains, and algebraic correspondences — through the notion of "mating" (combining two dynamical systems). Among the concrete results: Julia sets of anti-rational maps can be realised as limit sets of reflection groups, and the parameter spaces of Schwarz reflections are in explicit correspondence with parameter spaces of anti-rational maps.

Techniques and tools. David surgery (a generalisation of quasiconformal surgery that allows sets of positive area), quasi-conformal straightening, mating theory for rational maps, and the combinatorial theory of Hubbard trees and laminations for anti-rational maps. Kleinian group methods (limit sets, Poincaré series, Patterson–Sullivan theory) are imported from hyperbolic geometry.

Proof idea. Being a survey, the paper collects and organises results from the authors' earlier work and the broader community. The conceptual thread is that each of the four dynamical classes arises as a "limit" or "degeneration" of another via mating, and that Schwarz reflections serve as a unifying interpolation object. Key analytic results are the David surgery constructions that realise combinatorial data (e.g., the topology of a Julia set) as an actual conformal dynamical system.

Authors: Xian Dai, Gerhard Knieper

Abstract (original): In his seminal work on Teichmüller spaces, Thurston introduced the maximal stretch for a pair of hyperbolic metrics on a closed surface of genus and showed that the logarithm of this quantity induces an asymmetric metric in the Teichmüller space. He also showed that the subset of the surface on which the maximal stretch is attained is a geodesic lamination. In this paper, we define the maximal stretch analogously for closed manifolds equipped with Riemannian metrics of variable negative curvature and investigate the structure of the related Mather set on the unit tangent bundle. In contrast to the Teichmüller space, the Mather set may not be lifts of geodesic laminations in this broader setting. However, in our paper, we will discuss similar features shared by the Mather set with geodesic laminations. We also connect the study of the Mather set with the theory of best Lipschitz maps.

Key result. Thurston's maximal stretch (the Lipschitz constant of the optimal map between two hyperbolic surfaces) and his result that the maximally stretched set is a geodesic lamination are extended to closed Riemannian manifolds with variable negative curvature. In this broader setting, the Mather set (the generalisation of the maximally stretched locus) lives on the unit tangent bundle and may fail to be a lamination — but it shares several lamination-like properties. A connection to the theory of best Lipschitz maps (calibration-type arguments) is established.

Techniques and tools. Aubry–Mather theory from Lagrangian dynamics provides the framework (the "Mather set" terminology and the variational perspective on optimal transport). Riemannian geometry of negatively curved manifolds, geodesic flow ergodic theory, and the theory of calibrations (closed forms that certify Lipschitz optimality) are the main tools. The connection to best Lipschitz maps draws on Thurston's original approach reinterpreted through calibration theory.

Proof idea. The maximal stretch is defined as the infimum of Lipschitz constants over all homotopy classes of maps, and the Mather set is the set of unit tangent vectors that are "maximally stretched" by the optimal Lipschitz map. The authors analyse the dynamics of the geodesic flow restricted to the Mather set, establishing its invariance and recurrence properties. For lamination-like behaviour, they show that the Mather set projects to a set in the manifold with certain transversality and regularity properties — weaker than a true lamination but analogous in the relevant dynamical sense.

Authors: Ilya Kapovich

Abstract (original): By the classic results of Fricke and Klein, for every word $w$ in the free group $F(a,b)$ there exists a unique integer trace polynomial such that for all . In this paper we study quantitative aspects of trace polynomials. We prove an exact formula for the leading homogeneous part of for every nontrivial cyclically reduced word . In particular, if is cyclically reduced over , where and , and if denotes the number of cyclic occurrences of the adjacent pair $rs$, then Consequently we obtain sharp general bounds for arbitrary $w$ and an exact formula for positive words . We also study for random positive words and for random freely reduced and cyclically reduced words. We also obtain explicit exponential upper bounds for the growth of the and norms of and exhibit examples with exponential coefficient growth at rate , where is the golden ratio. Finally, we give a deterministic algorithm which computes the fully expanded polynomial in time and space , and prove an output-size lower bound for all algorithms that output in fully expanded sparse binary form. As a consequence, -character equivalence for elements is decidable in time .

Key result. Exact formula for the degree of the trace polynomial in terms of the combinatorial structure of the word $w$: , where $n = |w|$ and counts cyclic adjacent-pair occurrences. This implies sharp bounds and resolves the polynomial degree in closed form. As a computational corollary, -character equivalence of two words is decidable in time.

Techniques and tools. The paper operates in the character variety of the free group $F(a,b)$. Trace polynomials go back to Fricke–Klein; their degrees are studied here using a combination of combinatorial group theory (cyclic word structure, adjacent pairs), generating-function methods for random word models, and polynomial norm estimates (Mahler measure type). The algorithm for computing uses matrix recursion and polynomial arithmetic; the lower bound is an output-size argument (the polynomial itself can be that large).

Proof idea. The degree formula is proved by tracking the leading-degree terms through the Fricke–Klein recursion: multiplying matrices $w(A,B)$ and applying the trace adds degree at most 1 per letter, but adjacent pairs $ab$ or $BA$ cause cancellations that reduce the degree by exactly 1 each. The leading homogeneous form is then identified explicitly. The probability results for random words use central-limit-type arguments on the random variable as the word length . Norm bounds follow from standard estimates on polynomial coefficients via the recursion.