Papers

We show that the Torelli group of a simply connected closed 5-manifold $M$, which is spin and has no 2-torsion elements in homology, is isomorphic to the bordism group ${\Omega}^{\mathrm{Spin}}_6(K(H_2(M), 2))$. For $H_2(M)$ with no 2- and 3-torsion we determine this bordism group, and give explicit constructions for the generators of the Torelli group. Furthermore, for the $g$-fold connected sum ${\#}^g(S^2 \times S^3)$ we completely determine its mapping class group. We apply our results to compute the stabilization and abelianization of the mapping class group of ${\#}^g(S^2 \times S^3)$, determine the group of isotopy classes of diffeomorphisms of $M$ that are homotopic to the identity, and study the embeddings of $S^3$ in $S^2 \times S^3$.

We show that for every sufficiently large genus $g$, there exists a closed hyperbolic surface $S_g$ with systole $\mathrm{sys}(S_g)\geq \log g-12\log\log g$. In particular, $$\liminf_{g\to \infty}\frac{\max\{\mathrm{sys}(S):S\in \mathcal{M}_g\}}{\log g}\geq 1,$$ improving the previously known bound $2/9$. This note is a continuation of our previous work on the diameter of finite covers arXiv:2608.12887, using the same framework of constant-twist pants decomposition to study systoles. The proof was developed by GPT-5.6 Sol through an extended discussion with the author.

We study real ideal points of $SL_2(\mathbb C)$-character varieties of knot exteriors containing essential Conway spheres. Under explicit deformation-theoretic hypotheses on the two complementary tangles, we show that essential Conway spheres are detected via Culler-Shalen theory with real ideal points; such ideal points are then deformed into arcs of $SL_2(\mathbb R)$ representations on both sides. We then compute the asymptotic behavior of the representations on these arcs and compute the translation numbers of their lifts to $\widetilde{PSL}_2(\mathbb R)$ representations. Combining this with a result of Gao, we conclude that for certain knots with essential Conway spheres, all sufficiently large positive and negative rational fillings have left-orderable fundamental group. This verifies the $L$-space conjecture for large-slope Dehn fillings of knots which were previously unknown in the literature. We verify the criteria for three infinite families of knots assembled from torus-trivial and twist-trivial tangles, then count the resulting real ideal points, determine the longitudinal translation numbers of all constructed branches, and identify exactly the zero-translation arcs. Computations for the $28$ distinct verified census exteriors listed in Table \ref{tab:instances} agree with the certified branches.

Poincaré duality is a classical theorem relating the homology and cohomology of closed oriented manifolds. This theorem has been extended to more general settings and to generalized (co)homology theories, including $K$-theory. In this paper, we construct an infinite-dimensional analogue of the $K$-theoretic Poincaré duality homomorphism. More precisely, for an infinite-dimensional Hilbert manifold $\mathcal{M}$, we construct a homomorphism $$K^{geo}_*(\mathcal{M})\to K_*(\mathcal{A}(\mathcal{M})),$$ where $K^{geo}_*(\mathcal{M})$ denotes the geometric $K$-homology of Baum and Douglas, and $\mathcal{A}(\mathcal{M})$ is a $C^*$-algebra associated to $\mathcal{M}$, based on a construction of Gong, Wu, and Yu. We also prove that the constructed homomorphism is non-trivial in certain cases.

We construct simply connected integral symplectic four-manifolds forming an exotic pair, with equal values of $c_1^2$, $c_2$, $c_1\mathbin{\cdot}[\omega]$, and $[\omega]^2$, but with symplectic Kodaira dimensions $-\infty$ and $2$. Since symplectic Kodaira dimension is invariant under Luttinger surgery, the two manifolds are not Luttinger-surgery equivalent. This gives a negative answer to Auroux's uniqueness question.

Let $M$ be a Weil-Petersson random hyperbolic surface of genus $g$, and let $\Gamma \subset \mathbb{Z}^{2g}$ be a lattice of prime index $q$. We study the distribution of primitive closed geodesics in homology classes mod $\Gamma$ in the large genus limit. Averaging over all lattices of index $q$, with $q \to \infty$, we compute all the centered moments of the corresponding weighted counting functions, and exhibit a transition between Poisson and Gaussian regimes (depending on whether $\frac{X}{q\log X}$, the expected number of primitive geodesics in a given homology class mod $\Gamma$, tends to $\lambda>0$ or $\infty$). We also study the unnormalized variance $G_M(X,\Gamma)$ of the counts among homology classes, and show that as $X \to \infty$, averaged over all lattices of prime index $q$, it is asymptotic to $X\log X$ in the large genus limit. These results are analogous to phenomena arising in the distribution of primes in arithmetic progressions.

We present a direct geometric construction which extends the Reshetikhin–Turaev TQFT to circles. We prove that this agrees with the generators-and-relations approach of Bartlett–Douglas–Schommer-Pries–Vicary, thus providing an alternative to the Cerf-theoretic part of their classification of once-extended 3-dimensional TQFTs, and deduce as a result that all once-extended 3-dimensional TQFTs arise via our construction.

Reformulated in terms of the asymptotic rank, a conjecture by Gromov predicts a linear isoperimetric filling inequality in all dimensions greater than or equal to the asymptotic rank of a Hadamard space, in contrast to the Euclidean-type nonlinear behavior below this threshold. We prove the predicted linear inequality for Hadamard spaces with finite asymptotic Nagata dimension and finite asymptotic rank. More precisely, every integral cycle of dimension at or above the asymptotic rank admits a filling whose mass is bounded linearly in the mass of the cycle. Our proof is based on a new self-improvement mechanism for Wenger's sub-Euclidean growth theorem. This approach upgrades the asymptotic rank-one result by Wenger and the recent asymptotic rank-two result by Lang, Stadler, and Urech from exponents arbitrarily close to one to the optimal linear exponent. Moreover, the result extends to arbitrary finite asymptotic ranks.

For surfaces of genus $g \geq 3$ with at most one boundary component, we give explicit generating sets for the kernel of the Birman-Craggs-Johnson homomorphism and the commutator subgroup of the Torelli group. As an application, we give a new proof of Johnson's calculation of the abelianization of the Torelli group.

We construct infinitely many new pairwise nondiffeomorphic smooth structures on infinitely many closed simply connected spin 4-manifolds with positive signature. Our construction builds on an infinite family of simply connected complex surfaces of general type due to Roulleau and Urzúa that populate points arbitrarily near the Bogomolov-Miyaoka-Yau line in the complex geography plane. We also discuss how the conjectural symplectic Bogomolov-Miyaoka-Yau inequality implies that our results are close to being optimal.

Multivariable Alexander theory produces Fitting ideals without preferred generators. After fixing a coefficient field and a monomial order, we represent these ideals by reduced Gröbner bases of their polynomial contractions, obtaining Alexander–Gröbner invariants of links. Over \(\mathbb Q\), for the maximal free-abelian coefficient system of a connected compact oriented \(3\)-manifold whose nonempty boundary is a disjoint union of tori, we deduce determinant-divisor reciprocity from Blanchfield duality. We compute these invariants for torus links and low-crossing links, and we determine \(\AG_2\) for a two-component pretzel family.

We construct the half-signature bordism bicategory $\mathbf{Bord}_{1,2,3}^\mathrm{sig/2}$ as a central extension of the 3-dimensional oriented bordism bicategory by $\mathbb{Z}$. This is an index $2$ subextension of the signature bordism bicategory. Following methods of Bartlett–Douglas–Schommer-Pries–Vicary, we construct a presentation of this bicategory and hence show that its linear representations are classified exactly by modular tensor categories.

We establish a grading bound for colored $\mathfrak{gl}_N$ link homology that can be read off from any link diagram. In the context of contact geometry, this gives a bound on the Thurston–Bennequin number of a Legendrian link from $\mathfrak{gl}_N$ link homology, in the spirit of Ng.

A typical example of a quandle is the conjugation quandle. A quandle is said to be admissible if it is isomorphic to a conjugation quandle. We study the admissibility problem for quandles, that is, determining whether a given quandle is admissible. In particular, we focus on generalized Alexander quandles, which are groups equipped with quandle structures defined by group automorphisms. In this paper, we provide a sufficient condition for admissibility in terms of geometric properties of quandles: namely, if every antipodal set in each algebraically connected component consists of a single point, then the quandle is admissible. This result generalizes previous results on generalized Alexander quandles.

Our main result is a computation of the full link Floer complex of the Borromean rings. The Borromean rings are well known to be an L-space link. We prove that the full link Floer complex is not a free-resolution of its homology, in contrast to the situation for L-space knots, plumbed L-space links, and two-component L-space links. We also describe the link involution of the Borromean rings, up to a minor ambiguity. The proof goes by way of studying some general techniques about $A_\infty$-deformations of L-space link Floer complexes. Our techniques give a simple criterion for when an L-space link must have formal link Floer complex, which reproves the existing formality results for one- and two-component L-space links.

We apply the $d$-invariants of Ozsváth-Szabó and a theorem due to Brendan Owens to determine the unknotting numbers of 194 alternating knots with 13 or fewer crossings.

We solve the three-dimensional symplectization question for several examples. These include small Seifert fibered $L$-spaces, a family of Whitehead-link surgery $L$-spaces, $S^1\times S^2$, certain surgeries on the figure-eight knot, explicit families of Brieskorn spheres, and arbitrary finite connected sums mixing these manifolds. In particular, the result covers the Weeks manifold and infinitely many hyperbolic $L$-spaces. The main symplectic step is to construct Liouville cobordisms in both directions from a symplectomorphism of symplectizations. The resulting preservation of strong fillability and of the vanishing of the monopole Floer contact invariant, together with the comparison with the Heegaard Floer contact invariant, connected-sum formulas, and classification results for tight contact structures, proves the stated rigidity results.

Cheng introduces the multivariate writhe polynomial, leaving a question regarding the characterization of this invariant. In this paper, we resolve this question by proving that a Laurent polynomial $f$ with integer coefficients can be realized as the multivariate writhe polynomial of some multi-virtual knot if and only if $f(1,\dots,1)=0$ and $\left. \frac{d}{dt} f(t, \dots, t) \right\vert{}_{t=1} = 0$.

This article arose out of notes for a CIME summer school mini-course in Cetraro. In it we develop differential-geometric constructions on the moduli space ${\cal M}_{g,n}^{\rm spin}$ of smooth spin curves, with particular emphasis on their interpretation in hyperbolic geometry. We relate the locally constant sheaf arising from a hyperbolic spin structure to the corresponding holomorphic sheaf, extending this known relationship to spin curves with Ramond marked points. We use the hyperbolic description to construct characteristic forms and spin measures on moduli spaces of hyperbolic surfaces with geodesic boundary, providing a framework for applying Teichmüller-theoretic methods to their volumes and volume recursions. Finally, we study separately the even and odd components of spin moduli space and find symmetries between their volume contributions which are not apparent from the contributions of individual boundary strata.

For an elliptically fibered 4-manifold $\pi\colon M\to B$, we introduce a subgroup of the smooth mapping class group of $M$ with explicit geometric representatives: the smooth narrow Mordell-Weil group $MW_0(\pi)$. Leveraging Kodaira's classification of singular fibers, we construct a natural isomorphism between $MW_0(\pi)$ and the orthogonal complement of the trivial lattice in $H^2(M)$. This identification parallels its holomorphic counterpart. This paper generalizes work of Farb and Looijenga, who defined the smooth Mordell-Weil group and computed it for fibrations over a sphere with only nodal fibers. We give an explicit formula for the rank of $MW_0(\pi)$ depending only on the genus of the base curve and the types of singular fibers of $\pi$. As a corollary, we show that rational elliptic surfaces are the only elliptic surfaces with holomorphic and smooth Mordell-Weil groups of the same rank.

Suppose that $f : M \rightarrow B$ is a normal $(q-1)$-smoothing of a $2q$-manifold $M$ over some $(B,\xi)$, where $q$ is even and $B$ is simply-connected. A diffeomorphism of $M$ over $B$ induces an automorphism of the extended intersection form (or “Q-form") of $f$, which consists of the intersection form of $M$ and the induced homomorphism $H_q(f)$. Assuming that $H_q(B)$ is free, we determine precisely which automorphisms can be realised by such diffeomorphisms. In particular, if $H_{q-1}(B)$ is also free, then we show that every automorphism can be realised. These results are obtained by studying a “two-sided" version of the extended surgery obstruction, which was introduced in earlier work of the author. As an application, we show that for a complex $q$-dimensional complete intersection, with $q > 2$ even, every automorphism of the cohomology ring is realised by a diffeomorphism.

Hyperbolic surfaces are a fundamental object in mathematics and play an increasingly important role in computational geometry and topology. A key ingredient in the design of efficient algorithms on such surfaces is the availability of a geometric discretization of controlled complexity. In this paper, we present the first algorithm for constructing e-nets on hyperbolic surfaces starting from a fundamental polygon representation. Our approach is based on Delaunay refinement and relies on maintaining Delaunay triangulations through edge flips. The size of an e-net cannot be bounded solely as a function of the genus because of the presence of arbitrarily long collars around short geodesics. To overcome this difficulty, we introduce the notion of a pseudo e-net, which decomposes the surface into e-thin cylinders together with a Delaunay triangulation over an e-net of the remaining thick part. As applications, we obtain algorithms for computing the length spectrum of an e-thick hyperbolic surface and for computing the systole from a pseudo log(sqrt(2))-net. These results demonstrate that Delaunay-based discretizations provide a practical and versatile framework for algorithmic computations on hyperbolic surfaces.

Kwasik and Schultz constructed two-ended open $4$-manifolds which satisfy the usual finiteness and stability conditions at infinity but do not admit arbitrarily small $1$-neighborhoods. In particular, neither end is collarable, so the manifolds are not completable. We show that the open manifolds associated to their non-desuspendable $C_2$-actions nevertheless admit finite-dimensional compact ANR $\mathcal{Z}$-compactifications. Consequently, there exists a $\mathcal{Z}$-compactifiable open $4$-manifold which is not pseudo-collarable. This answers a question of Guilbault and Tinsley.

In this short note, we exhibit a Legendrian simple knot that is not detected by Khovanov homology. This disproves a conjecture of Chernov and Maguire.

We develop a systematic method for computing traces of products of Möbius transformations associated with oriented geodesics on a hyperbolic surface. The method is based on a normalization of matrices in ${SL}_2(\mathbb R)$ which expresses trace identities in terms of hyperbolic lengths, intersection angles, and signed distances along geodesics. Using these trace computations together with Goldman's description of the Atiyah-Bott-Goldman symplectic form on the character variety, we derive explicit geometric formulas for the Poisson brackets of trace functions associated with closed geodesics. More precisely, we express the Poisson bracket of two trace functions and the iterated Poisson bracket of three trace functions in terms of the hyperbolic lengths of the corresponding geodesics, their intersection angles, and the signed distances between intersection points. The results naturally lead to a unified perspective for revisiting Wolpert's cosine and sine formulas and deriving new proofs of them.

For an oriented link diagram $D$, the Khovanov cochain complex carries a canonical Hermitian inner product that defines a combinatorial Hodge Laplacian. Its kernel is naturally isomorphic to rational Khovanov cohomology, while its positive spectrum depends on the chosen diagram. Building on the numerical work of Jones and Wei, we develop a structural study of this diagram-dependent higher spectrum. In the minimal and maximal cube degrees, we identify the Khovanov Laplacian with signless Laplacians of explicit weighted graphs, up to diagonal sign conjugation in maximal degree. The graph model describes rational Khovanov classes and harmonic representatives through its bipartite components and gives inverse-polynomial gap bounds in fixed-width bands near the extremal $q$-degrees when the corresponding rational cohomology vanishes. It also yields exact $\Theta(N^{-2})$ bidegree gaps for twisted unknots, odd $(2,N)$-torus knots, and even twist knots. The exact gap calculations are motivated in part by the spectral-resolution requirement in a recent quantum algorithm for Khovanov homology. We also prove a finite-dimensional analytic–combinatorial torsion correspondence for the Khovanov complex showing that an alternating product of nonzero Laplacian pseudodeterminants recovers integral Khovanov torsion in rationally acyclic $q$-degrees and differs from it by an explicit regulator in general. Finally, we transfer Lee's deformation to a filtered differential on the harmonic Khovanov subspace, giving a canonical harmonic model for the Lee spectral sequence.

A Lefschetz fibration is a smooth 4-manifold admitting a surface bundle structure over a surface except at finitely many singular fibers, whose singularities are only of nodal type. From the structure of the singular fibers, the monodromy of each singular fiber is given by a right-handed Dehn twist along a curve, called a vanishing cycle, in the fiber. Collecting all monodromy data from the singular fibers, we obtain a monodromy factorization into right-handed Dehn twists, which completely determines the Lefschetz fibration. In this paper, we construct a fibration with one singular fiber whose monodromy is periodic (that is, the monodromy homeomorphism is isotopic to a periodic map). Following the idea of Matsumoto, we give a splitting of the singular fiber into Lefschetz fibers and determine their vanishing cycles for a collection of periodic monodromies. We describe the construction of the splitting singular fibers and the procedure for reading vanishing cycles using two branched covering structures of the fibers. We also give splittings of singular fibers into Lefschetz fibers related to a family of periodic actions and determine their vanishing cycles.

Let $(M^3, g_{\mathrm{hyp}})$ be a closed oriented hyperbolic three-manifold normalized so that $\operatorname{sec}_{g_{\mathrm{hyp}}} \equiv -1$. We prove a sharp lower bound for the volume of hypersurfaces in $M^3 \times \mathbb{S}^1$ representing the slice class $[M^3 \times \{ \mathrm{pt} \}] \in H_3(M^3 \times \mathbb{S}^1; \mathbb{Z})$, and we classify the equality case. If $g$ is a smooth Riemannian metric on $M^3 \times \mathbb{S}^1$ with the scalar curvature $\operatorname{Sc}_g \geq -6$, then every closed embedded hypersurface $\Sigma$ representing the slice class $[M^3\times\{\mathrm{pt}\}]$ satisfies $\operatorname{vol}_g(\Sigma) \geq \operatorname{vol}_{g_{\mathrm{hyp}}}(M^3)$. The bound is attained by the product metric $g_{\mathrm{hyp}}+h$, with $h$ any metric on $\mathbb{S}^1$. Conversely, if equality holds for some $\Sigma$, then up to a diffeomorphism preserving the slice class, $g=g_{\mathrm{hyp}}+h$ and $\Sigma = M^3 \times \{\mathrm{pt}\}$.

Classifying link types via their n-bridge positions requires navigating vast Hilden double coset classes H2n\B2n/H2n. We provide a restriction on this by proving that every Hilden double coset admits a positive braid representative whose Dehornoy minimum is explicitly bounded. The proof relies on the key observation that $\Delta\in H_{2n}$, which allows us to utilize the Garside left-greedy normal form to remove all powers of $\Delta$ from any given braid representative. We provide an explicit upper bound on its Dehornoy minimum in terms of the Garside length of the initial presentation. This bounded representative restricts the algebraic space of plat presentations, providing a potential computational tool for studying bridge isotopy classes and offering a constructive step toward algebraic approaches to the Bridge Finiteness Conjecture.

We consider closed connected oriented three-manifolds equipped with positive cooriented contact structures. We prove that a symplectomorphism between their symplectizations forces the existence of an orientation-preserving diffeomorphism between the underlying three-manifolds under which the two contact structures are homotopic as oriented plane fields. Combining this formal rigidity with the classification of overtwisted contact structures, we show that two closed overtwisted contact three-manifolds have symplectomorphic symplectizations if and only if they are contactomorphic. Thus, we resolve the three-dimensional symplectization question when both contact structures are overtwisted.