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Wednesday, 4 November 2026: 12 new papers, 4 cross-lists and 6 replacements in Geometric Topology (math.GT), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We give a combinatorial description of the Heegaard--Floer link-cobordism maps associated with pointed triple grid diagrams in the complement of three $4$-balls in complex projective plane with reverse orientation. Every contributing index-zero holomorphic triangle decomposes into Euclidean triangular sheets. We characterize the compatible collections of sheets that contribute by an explicit geometric separation criterion, expressed as a lattice-point test, and prove that each has mod-\(2\) count one. Individual sheets may wrap around the torus in the unblocked theory, whereas they are embedded in the blocked theory. In each fixed \(\SpinC\) structure, the map is determined by a finite enumeration. A difference number computed from the triangular regions determines the \(\SpinC\) structure of a Whitney triangle. Together with the topology of the associated surface, this gives an explicit formula for the absolute bigrading of each summand of the cobordism map. After basepoint reduction, the formula determines the Laurent monomial representing any nonzero map on the rank-one torsion-free two-variable knot Floer modules, yielding vanishing and ideal-containment criteria. We apply the construction to an affine family of grid links \(K(n;p,s)\), obtaining explicit cobordisms involving infinite families of torus knots and links, together with closed formulas for the bigradings of their distinguished grid states.
We study the Farrell--Jones conjecture for graph products of pairs and for fundamental groups of polyhedral products. We introduce a pair-deletion decomposition and a local notion of FJ-admissibility, and prove a vertexwise combination theorem for graph products of pairs. The criterion applies, in particular, to ordinary graph products, free-factor pairs, suitable normal pairs, and classes controlled by acylindrical Bass--Serre actions. For polyhedral products, we combine these results with their description by relative graph products and graph products of pairs. We also obtain bounds and exact computations for amenable category, together with applications to simplicial volume and Borel rigidity of closed aspherical polyhedral-product manifolds.
We present discrete, faithful and purely loxodromic representations of a closed surface group of genus 431, and of a free group, into the group of isometries of a finite-dimensional real-hyperbolic space for which no Cannon--Thurston map exists. The construction combines three ingredients: the small-cancellation strategy employed by Baker--Riley to construct a subgroup of an ambient hyperbolic group with no Cannon--Thurston map, the realization of these groups by octagon complexes, and a theorem of Desgroseilliers--Haglund that supplies a convex cocompact faithful representation of the ambient hyperbolic group on a real-hyperbolic space. These examples admit ray Cannon--Thurston maps, as defined by Halder, Mahan Mj and Sardar.
The geometric dimension of a nonempty compact PL manifold $N$ is the smallest nonnegative integer $d$ for which $N$ admits a handle decomposition with all handles of index at most $d$. A PL filling of a compact PL manifold is a decomposition into compact codimension $0$ PL submanifolds, called sectors, with pairwise disjoint interiors and intersections described by standard local models. We advance Singhof's program of simplifying the sectors of a filling and their intersections. Specifically, for integers $m, \ell\ge 1$ and nonnegative integers $a_1\le\cdots\le a_\ell$, we prove that every closed PL $m$-manifold admits a filling into $\ell$ sectors whose nonempty $r$-fold intersections have geometric dimension at most $a_r$ for each $1\le r\le \ell$ if and only if \[ a_{\lambda_1}+\cdots+a_{\lambda_q}\ge m-\ell+1 \] for every partition $\lambda_1+\cdots+\lambda_q=\ell$ into positive integers. Under the same inequalities, any prescribed $\ell$-sector filling of the boundary of a compact PL $m$-manifold extends to a filling with the corresponding relative handle index bounds. Necessity follows from the cup-product obstructions of Courte, Moussard, Ren, and Zhou. To prove sufficiency, we develop compatible handle trades that simplify intersections in increasing order of depth. For nondecreasing tuples of nonnegative integers, our theorem resolves the PL classification problem posed by Courte, Moussard, Ren, and Zhou. In particular, we obtain the bound $(2,2,4)$ for $3$-sector fillings in dimension $6$ and improve the $5$-sector bound $(1,2,3,3,4)$ of Rubinstein and Tillmann to $(1,2,2,3,4)$ in dimension $8$.
We compare linking-form bounds with the smooth and locally flat nonorientable 4-genera of low-crossing knots. For every prime knot with at most 13 crossings whose Witt-level linking-form bound is at least 2, this bound equals both genera. We also exhibit 41 prime knots with 12 or 13 crossings that bound locally flat M\"obius bands but no smooth ones, and whose smooth nonorientable 4-genus is 2. Our methods combine linking forms, Heegaard Floer correction terms and definite-lattice obstructions with explicit band constructions. They determine 3,300 of the 3,525 values left unknown in KnotInfo on 2 November 2026; the remaining 225 knots have range $[1,2]$. We obtain new lower bounds of 2 for 2,102 knots with 14 crossings and correct previously reported values for three knots with 11 crossings.
For closed 4-manifolds $X_1, X_2$, we give a sufficient condition, in terms of the Seiberg-Witten invariants of $X_i$, for the Dehn twist on $X_1\#X_2$ along the neck to be smoothly nontrivial. This provides a broad generalization of the result of Kronheimer-Mrowka for $K3\#K3$. We also show that the boundary Dehn twists on the punctured connected sums of many pairs of elliptic surfaces are smoothly nontrivial. We also detect new examples of exotic Dehn twists along Seifert fibered 3-manifolds in elliptic surfaces. As a consequence, we exhibit a new phenomenon in which "exotic diffeomorphisms detect exotic smooth structures on 4-manifolds": the triviality of the boundary Dehn twist characterizes the standard nucleus $N(n)$ among the nuclei $N(n)_{p,q}$ homeomorphic to it, provided that $n\equiv0 \bmod 4$. These results are proved by computing the families Bauer-Furuta invariant for a new framing of the families Dirac index.
We define a Morse homology of the foliation which is defined by a general nonsingular Morse-Smale flow, applying the framework of the Conley index theory. In fact, we define a chain and a boundary homomorphism, therefore, a cochain and a cohomology are obtained as well. Furthermore, we prove a duality theorem on the homology and the cohomology.
We prove that a knot $K \subset S^3$ is the unknot if and only if the $\mathbb{Z}/2$-equivariant Seiberg--Witten Floer cohomology of the double cover branched along the $(2, 1)$-cable of $K$ has no $\mathbb{F}_2[Q]$-torsion. The proof combines the Thom--Gysin exact sequence, the Montesinos trick, and the genus bound for $L$-space surgeries.
A theorem from Wolpert bounds the Weil--Petersson distance from a hyperbolic surface to a maximally noded surface in terms of the lengths of the underlying pants decomposition. We prove analogues for punctured surfaces both in terms of the lambda-lengths and in terms of the shearing coordinates of an ideal triangulation.
Let X be a closed oriented surface of genus at least 2 and P a finite subset of X. We bound the action of the mapping class group of (X,P) on the rational cohomology groups of the ordered configuration spaces F_n(X-P) of X-P. In particular, we show that, when #P < 2, the diagonal action of the mapping class group on the direct sum of the H^n(F_n(X-P)) is faithful, and determine its kernel when #P > 1.
This paper studies a subgroup of the mapping class group of an elliptically fibered $4$-manifold $\pi: M\to B$, inspired by number theory. The smooth Mordell-Weil group $\mathrm{MW}(\pi)$ is the group of isotopy classes of smooth sections of $\pi$. We construct a smooth analog of the Shioda-Tate isomorphism, identifying $\mathrm{MW(\pi)}$ with the quotient of $H^2(M;\mathbb{Z})$ by the trivial lattice of $\pi$. This shows that $\mathrm{MW}(\pi)$ can be computed explicitly from $H^2(M;\mathbb{Z})$. As an application, we show that the natural inclusion of the holomorphic Mordell-Weil group into the smooth one is an isomorphism if and only if $M$ is a rational elliptic surface. As another application, we show that $\mathrm{MW}(\pi)$ satisfies a strong version of Nielsen realization: any torsion element of $\mathrm{MW}(\pi)$ is represented by a holomorphic translation of $M$ of the same order.
We introduce a version of convex surface theory for immersed ribbon surfaces with Legendrian boundary. As an application, we establish a special case of K3: Problem 1.70 on sharpness in the slice-Bennequin inequality implying quasipositivity, and a special case of a conjecture of Cornwell, Ng, and Sivek on characterizing Lagrangian sliceness. Specifically, we prove that if $K$ is a knot that bounds a "star-shaped" ribbon disk and satisfies $\overline{\mathrm{sl}}(K) = -1$ (resp.\ $\overline{\mathrm{tb}}(K) = -1$), then $K$ is quasipositive (resp.\ Lagrangian slice). In fact, we prove these statements for any fixed transverse (resp.\ Legendrian) representative.
The ensemble average of Narain conformal field theories over their moduli space is a real-analytic Eisenstein series, whose Poincar\'e expansion can be interpreted as a sum over lens spaces weighted by the partition functions of Abelian Chern-Simons theory. In this paper we extend this correspondence from lens spaces to Seifert manifolds, which gives an example of a restricted sum over topologies of the kind that arises in three-dimensional quantum gravity. A Seifert manifold is obtained from a fixed core, a punctured surface times a circle, by gluing solid tori to its boundary tori, and we place a copy of the Narain theory on each of these tori. The ensemble average supplies the sum over the solid tori, or more precisely over marked presentations with fixed base genus, number of fibers and integer Seifert invariant. The resulting sum converges absolutely and is a polynomial in the components of the vector-valued Eisenstein series. When all the boundary tori share a single modulus, the average acquires connected corrections, which we identify with wormholes gluing the core to itself; for two boundary tori, the terms with rational Lagrangian sublattices of the Siegel-Eisenstein series are realized by wormholes containing an exceptional fiber. After reduction along the fiber, the construction closely parallels the two-dimensional topological gravity of Marolf and Maxfield, with the base of the fibration playing the role of their spacetime. We also discuss Dehn fillings of 3-manifolds with torus boundary, non-orientable bases, and the surgery formula. Our results suggest that modularity could be an organizing principle in the summation of geometries in quantum gravity.
We survey the recent development of spinal open books and nearly Lefschetz fibrations. The focus is on explaining past and new results regarding the detection of symplectic fillability and classification of symplectic fillings of contact 3-manifolds in light of the new machinery. We give many examples and a list of open questions. This can be considered as an additional chapter to the lecture notes by Ozbacgi and Wendl.
Let $F$ be a finitely generated free group. We prove that the centraliser of any element of $\mathrm{Out}(F)$ or $\mathrm{Aut}(F)$ is of type $\rm{VF}$. We also prove that the (outer) automorphism group of any free-by-cyclic group $M=F\rtimes_\varphi\mathbb{Z}$ is of type $\rm{VF}$, and so are the McCool subgroups of $\mathrm{Out}(M)$. This relies on the construction of a new $\mathrm{Out}(M)$-invariant splitting of $M$, that enables us to reduce the proof of the above statements to the case where the outer class of $\varphi$ has finite order.
In this paper we begin developing a theory of Morse hierarchies for finitely generated groups. Very roughly, a Morse hierarchy is a iterative splitting of a group over subgroups that do not contribute to the Morse boundary of the ambient group. We apply this theory to Coxeter groups, establishing new combinatorial criteria that distinguish certain Coxeter groups from other Coxeter groups as well as right-angled Artin groups up to quasi-isometry. We highlight the consequences for the right-angled case. To build the reader's intuition, the last section provides many examples of Morse hierarchies and their subtleties. Finally, there is an appendix by the second author and Vivian He that extends these ideas to the metrizable Cashen--Mackay topology on the Morse boundary.