3-manifold fibering decision problem is in NP
A certificate verifiable in polynomial time exists whenever a triangulated manifold fibers over the circle.
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Geometric Topology
Manifolds, orbifolds, polyhedra, cell complexes, foliations, geometric structures
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Geodesic flows gain integer periods via metric dictionary
Every continuous reparameterization is matched to a group metric; word metrics yield flows with all orbits of integer length.
· “A geometric correspondence for reparameterizations of geodesic flows”
KAT inequalities fix circle patterns on quasi-simplicial surfaces
The possible curvatures for triangulations with loops and multiple edges are exactly those meeting the inequalities on every subset of the-l
· “Circle Pattern Theorem for Quasi-simplicial Triangulated Surfaces”
nV is torsion-locally finite
Higher-dimensional Brin-Thompson groups control torsion yet embed the rationals in continuum many distinct ways.
Overtwisted contact 3-manifolds are rigid under symplectization
A symplectomorphism of symplectizations now forces a contactomorphism, settling the overtwisted case.
· “The three-dimensional symplectization question for overtwisted contact structures”
Two numbers determine similarity-pair boundary dynamics
λ and θ encode the boundary map; if a conjecture holds, cut points are read from walks in a graph.
Quiver mutations determine plane-curve monodromy
The mutation class of a malleable divide uniquely recovers the Alexander module and the topological type.
· “Monodromy of plane curve singularities and quiver mutation”
Boundary topology fixes the quasi-isometry class of Kleinian groups
Kleinian groups are quasi-isometric exactly when Bowditch boundaries agree on carpets and cut-point orders.
All but five holonomy signatures are realizable
A reduction from $4^{2g}$ cases to three, via 4-differentials, closes the quadrangulation existence question.
· “Which Holonomy Signatures Are Realizable? A Complete Answer for Closed Surfaces”
Four elements generate the Goeritz group for genus g splittings of S^3
Powell's conjecture is settled for g at least 3 using topological minimality of the surface.
· “A proof of Powell's conjecture on the Goeritz group of S³”
All (p,q)-pinwheel embeddings in B_{p,q} are Hamiltonian isotopic
The nearby Lagrangian conjecture holds for these singular Lagrangians because the symplectomorphism group is generated by a single twist.
Brieskorn spheres block rational homology ball fillings
Correction-term and torsion obstructions rule out such fillings for all n=3 cases and most higher-n cases, confirming Gompf conjectures.
· “Brieskorn spheres and rational homology ball symplectic fillings”
Trace map for character varieties is never injective
For every n, explicit nonzero curve combinations yield trace functions vanishing on all surface-group maps to GL_n.
· “Non-injectivity of the trace map for character varieties”
The authors construct an irreducible embedded real projective plane inside the 4-sphere
An irreducible embedded projective plane is constructed in S^4, countering the Kinoshita conjecture via a peripheral map with kernel of…
A researcher has built a map connecting the topology of infinite-dimensional manifolds to…
For Hilbert manifolds with bounded geometry, the paper constructs a homomorphism from geometric K-homology to the K-theory of the…
· “Geometric K-homology and operator K-theory for Hilbert manifolds”
Four families generate the kernel of the BCJ map
The generating sets also describe the commutator subgroup and recover Johnson's abelianization of the Torelli group.
Spin 4-manifold geography bound improves to the 4/5 power
Surfaces hugging the BMY line cut the threshold from a 6/7 to a 4/5 power.
· “Slope Inequalities for the Geography Problem of Spin Symplectic 4-Manifolds”
Lasagna modules inherit Khovanov-Floer spectral sequences
On 2-handlebodies, taking homology of a lasagna module page gives the next page, yielding rank inequalities.
Every closed orientable 3-manifold has stably exotic fillings
The same 3-manifold can be filled homeomorphically yet not stably diffeomorphically; only a two-sided RP^2 forbids this.
This paper classifies positively curved spaces with a strong fixed-point symmetry action
Fixed point homogeneous positively curved orbifolds are, up to equivariant diffeomorphism, spheres, the Cayley plane, or weighted complex…
· “Fixed Point Homogeneous Orbifolds with Positive Sectional Curvature”
Spin-parity volumes of minimal strata now have a closed formula
The even-minus-odd Masur-Veech volume reduces to Hodge integrals via a recursion on spin-parity square cones.
Zeeman conjecture reduces to standard polyhedra
If any contractible 2-polyhedron fails ZCC, a standard one fails too — tying ZCC to the Andrews–Curtis conjecture.
· “On Zeeman's collapsibility conjecture for non-standard polyhedra”
For a class of 3-manifolds with an odd symmetry
Real Seiberg-Witten Floer homotopy types of Seifert 3-manifolds with odd involutions are shown to be equivalent to a suspension of C_+ by…
Every Olshanskii epimorphism is standard
High-genus finite-cover epimorphisms from a 1989 construction are proven equivalent to the standard splitting of S^3.
· “Equations in Products of Free Groups and 3-Manifold Groups II: Olshanskii Epimorphisms”
Volumes of flippered moduli spaces are finite and recursive
Three recursions—neck, disk, crown—reduce every volume to explicit integrals and closed formulas.
· “Flippered hyperbolic surfaces and renormalized volumes of their moduli spaces I”
Polyhedral surfaces get a discrete uniformization theorem
Every hyperbolic polyhedral surface maps to a standard model; Euclidean and spherical cases need one bound.
Teichmüller stars reach beyond zero sets
Thurston stars equal zero sets; Teichmüller stars equal two-step zero sets.
Fibered 5-manifold boundaries extend to relative 4-sections
A higher-dimensional analogue of Heegaard splittings now reaches 5-manifolds, with diagrams and gluing.
· “Relative multisections of higher-dimensional manifolds with boundary”
Explicit relations kill all generators in two exotic 4-manifolds
The full presentations prove the reverse-engineered manifolds are simply connected and match the standard types.
Third Johnson image is the Morita trace kernel from genus 6
For surfaces with one boundary, a single trace zero decides whether a derivation lies in the image.
3D link symmetry shrinks to a planar graph's automorphisms
For simple Hopf handlebody-links, 3D symmetry equals plane-graph symmetry; every finite subgroup of O(3) is realized.
· “Hopf handlebody-links: their symmetry and classification”
Flat arrangement polynomials are always trapezoidal
New polytope integral settles trapezoidality for every flat arrangement and every Eulerian digraph.
Vanishing H2 governs whether an aspherical 4D-pair is decomposable
PD4-pairs assemble from balls and algebraic pieces exactly when a group-theoretic cohomology vanishes
Eigenfunctions beat the classic sup-norm scale on symmetric spaces
A logarithmic saving (log T)^{-r/2} on thick quotients; strict o(T^{(n-r)/2}) locally.
6-valent vertex categorified by foams for gl4 and gl5
For N=4,5 the hexagon web is a direct sum of indecomposable foam objects; general N is conjectured.
· “6-valent vertex in the mathfrak{gl}_N web category and its categorification”
Hamilton's theorem is now machine-proved with zero custom axioms
Every closed 3-manifold with positive Ricci curvature is a spherical space form, now verified in the Lean assistant.
· “A Lean Formalization of Hamilton's Three-Manifold Theorem”
4-manifold invariant distinguishes phantom homotopy classes
Extends the known fusion-2-category state sum and distinguishes homotopy classes invisible to homotopy groups.
· “Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds”
r-coverings set lower bounds for knotoid moves
Virtualizing crossings by index preserves homotopy and shrinks Gordian distance predictably.
· “The r-coverings and local moves of planar virtual knotoids”
Doubling a clean knot rules out every SU(2)-abelian surgery
The result pins the status of every non-torus knot with at most nine crossings except two.
Kills both pi-2 maps in a concrete 2-complex chain
A small 8-generator construction answers an open 2-complex question and also refutes the Lie-ring analogue.
Squares, hexagons, octagons deform monotonically to maximal surfaces
Explicit continuous paths raise the shortest saddle connection at every step, ending at equilateral-triangle tilings.
· “Systole Increasing Deformations to Maximal Translation Surfaces”
The entropy spectrum of a hyperbolic surface is reverse well-ordered
Subsurface entropies accumulate exactly where the complement still has room for a curve, with a real gap below 1.
Every nonsplittable determinant-zero link has a 3-color Dehn diagram
Region-colorings need three colors for nonsplittable links; arc-colorings need four.
· “The minimum number of Dehn mathbb Z-colors of any nonsplittable mathbb Z-colorable link is three”
Tangent-sphere knots use fewer beads than any FCC lattice embedding
Explicit examples give every prime knot through seven crossings one fewer sphere than the lattice, and 8_1 two fewer.
· “Pearl necklace knots with fewer vertices than an equivalent FCC lattice knot”
Every non-orientable surface of genus 3+ gets a minimal filling pair
A fat-graph construction yields the fewest possible intersections, and orbit counts grow super-exponentially with genus.
q-series exponents explain 99.9% of d-invariant variance
On Seifert homology spheres, a short exponent prefix plus Dedekind sums gives near-exact correction-term recovery.
The D∞ subfactor is just the Brauer algebra at δ=2
It gives a diagrammatic calculus of virtual crossings for the D∞ principal graph, over any ring with 1/2.
· “The Brauer category mathcal{B}(2) has principal graph D_infty”
Every polynomial in z^2 is a Legendrian ruling polynomial
The paper constructs a Legendrian link for each such polynomial, completing the classification.
Exceptional eigenvalue counts decay polynomially on congruence covers
Settles the limit multiplicity conjecture for thin hyperbolic manifolds, with a geometric rate from shadow returns.
Equivariant sphere-to-Stiefel degrees collapse to one residue class
When the Hurewicz index is 1, every equivariant map has degree congruent to a binomial coefficient mod m, or none exist.
· “On the degrees of equivariant maps from spheres to complex Stiefel manifolds”
State-sum invariant recovers hyperbolic volume in cusp limit
The limit ties quantum state sums to classical geometry for knot and cusp complements.
· “mathrm {U}_(qtilde q)mathfrak{sl}(2;mathbb R) Turaev-Viro invariants for cusped 3-manifolds”
For p=2..8, Pontryagin surfaces are limit sets in H^5
These 2D boundaries need five dimensions; the paper proves dimension four is impossible for them.
· “Discrete embeddings of hyperbolic groups with Pontryagin-surface boundaries”
Two bilinear forms decide when real Heegaard Floer homology lifts to Z
The obstruction is a triple cup product with the branching monodromy, computed on twisted cohomology before any Floer theory.
Braid-kernel rule extends from surfaces to all complexes
Whether the kernel of the strand map has Goldberg's form is decided by a single tree condition on the complex.
· “Extending Goldberg's Exact Sequence to Braid Groups of Graphs and Simplicial Complexes”
Tetrahedron is the smallest equiangular hyperbolic polyhedron
For every angle between π/3 and arccos(1/3), no other equal-angle polyhedron has volume lower.
· “Minimal Equiangular Hyperbolic Polyhedra in the Tetrahedral Range”
4D torus trick proves crossing numbers add for all knots
A new equality between crossing number and chord index on spun ribbon torus-links settles connected-sum additivity.
A closed leaf fully decides a ray's limit type
Closed leaves split the ray boundary into big horospherical versus all other cases.
A labelled link diagram's state sum equals the Turaev–Viro invariant
Finite sum over edge and region colours of a branched-covering diagram reproduces the quantum invariant of the covered 3-manifold.
· “Reconstructing Turaev-Viro Invariants from Four-Fold Simple Branched Covering Presentations”
Every hyperbolic surface has finite covers of near-minimal diameter
For any closed hyperbolic surface, finite covers alone attain the log-genus diameter bound up to a 1+o(1) factor.
Unique incompressible surfaces trigger skein torsion at every genus
0-surgery on double twist knots gives one incompressible surface and cyclotomic torsion.
Explicit order-60 pairs give four groups infinite Nielsen classes
Explicit generating pairs stay distinct until a trivial coordinate is added.
· “Nielsen classes in outer automorphism groups of free groups”
Resolution graphs repeat with periods 10
Three infinite singularity families have compactified graphs fixed by the exponent mod 10, 6, or 8, with software to check them.
· “Construction and Periodicity of Minimal Resolution Graphs of Suspension Singularities”
Infinite sequence of Dehn fillings gets spherical CR structure
Uniformizing groups exist for every n≥5, turning Dehn fillings into spherical CR manifolds.
· “Spherical CR uniformizations of a sequence of hyperbolic 3-manifolds”
Finitely many singular-orbit links per closed atoroidal 3-manifold
Every pseudo-Anosov flow on a fixed manifold must realize one of finitely many singular link types.
Odd Vassiliev invariants vanish at four loops
No odd Jacobi diagram survives AS and IHX through four loops, so knot inversion remains undetectable.
Higher-degree block spaces carry new SL(2,Z) representations
For S3 in characteristic 3, torus block spaces in degrees 1 and 2 mod 4 cannot be built from degree zero.
· “Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups II”
Non-algebraic Anosov flows exist in every dimension 4+8n
Fibrewise construction over a 4D example yields stable dimension 2+4n and unstable 1+4n.
SL(2,C) character varieties get log Calabi-Yau compactifications
Compact surfaces get dlt boundaries; punctured ones get log canonical ones, settling a folklore conjecture.
· “Log Calabi-Yau compactifications of SL(2,mathbb{C}) character varieties”
Asymptotic cones of free groups always contain 2-torsion
Using explicit words whose square is sublinear, the cone gets a non-trivial order-2 element.
A mathematics paper computes the rational homotopy effect of the isometry group of the…
The rational homotopy class of the F4-action on the Cayley plane is determined, showing F4 and F4/Spin(7) map isomorphically to the…
· “Homotopical Robustness of Isometries on the Cayley Plane”
Knot traces preserve ν and |ε|
Two Floer invariants now separate exotic pairs and rule out symplectic fillings.
· “Heegaard Floer knot trace invariants, exotic 4-manifolds, and symplectic obstructions”
New polynomial separates twisted knots the Q-polynomial cannot
The new invariant also yields a compact move-generating set and a Vassiliev invariant of order one.
Same boundary, same Chern class, different Stein structures
Same boundary contact form and first Chern class; fillings are neither symplectomorphic nor Weinstein homotopic.
Toric real loci are kaleidoscopes of moment polytopes
A simple parity count on facet normals decides orientability, and the Euler characteristic becomes a weighted face count.
Dim of measured lamination space is a normal-surface max
In Haken 3-manifolds, dim ML equals the max relative dimension of least-weight faces, proving the DGR22 conjecture.
· “Growth Rate of Essential Surfaces via Measured Laminations”
Numerical search finds J-holomorphic curves in non-integrable settings
Starting from random smooth maps, the pipeline recovers known holomorphic curves and follows them as the geometry deforms.
· “Searching for J-holomorphic curves via machine: first steps”
Skein-algebra presentation conjecture fails in genus 3 and up
Every candidate algebra has a zero-character; the true skein algebra has none, so no isomorphism exists.
· “A Disproof of Santharoubane's Conjecture on Presentations of Generic Skein Algebras”
Braid monodromy on cyclic covers equals the Burau representation
New explicit unitary model of the reduced multivariate Burau representation via Pochhammer and figure-eight contour monodromy.
· “Geometric Monodromy of Mixed Braid Groups and the Multivariate Burau Representation”
Rack type t annihilates the classifying map in all degrees
For connected racks, every higher homology class lands in t-torsion; without connectedness, t^(n-1) kills torsion classes.
Every spherical tensor category yields a finite-dimensional 3d TFT
A spanning-tree trick controls red loops, and skein modules bound the TFT's state spaces.
