math.GT daily digest: 8 submissions for 5 November 2026

New submissions

Non-Pseudo-Arithmetic Hyperbolic 4-Manifolds

• Authors: Alexander Kolpakov
• arXiv: 2611.05795
• Comments: 20 pages; GitHub repository at this https URL
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We give three explicit finite-volume hyperbolic Coxeter lattices in dimension four whose adjoint trace fields are not totally real. They are the reflection groups of the Ma--Zheng polytopes , , . The obstruction is contained in two cyclic products of the doubled unit-normal Gram matrix . Those products recover elements and $u$ satisfying $$ (s-1)u^2+2(s-3)u+s-7=0. $$

After the embedding , this quadratic has discriminant . The cyclic-product field therefore has a complex place. Vinberg's identification of that field with the adjoint trace field and the Emery--Mila total-reality obstruction imply that the three lattices are not pseudo-arithmetic. Torsion-free orientation-preserving finite-index subgroups yield orientable noncompact finite-volume hyperbolic $4$-manifolds with the same obstruction.

Exact computations check the census data and provide independent field and finite-volume certificates. A pinned Lean~4 development verifies the radical identity and the number-field obstruction. It also checks the lattice and manifold implications conditional on explicit geometric and literature hypotheses.

On the (in)finiteness of fixed characters in genus $3$

• Authors: Yixuan Fu
• arXiv: 2611.06357
• Comments: 19 pages
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG)

We construct a counterexample to the Putman--Wieland conjecture in genus $3$ and discuss some of its consequences.

3-manifold decision problems and Riemann hypotheses

• Authors: John A. Baldwin, Steven Sivek
• arXiv: 2611.06705
• Comments: 11 pages
• Subjects: Geometric Topology (math.GT)

Beginning with Kuperberg's work on unknot recognition, several fundamental decision problems in 3-manifold topology have been shown to be in coNP assuming the generalized Riemann hypothesis. In this note, we explain how recent developments around the quasi-Riemann hypothesis yield unconditional versions of these results, leading for example to the first unconditional proof that 3-sphere recognition is in coNP.

Chromatic Spherical and Renormalized Reshetikhin-Turaev Invariants for Drinfeld Centers

• Authors: Nicolas Bridges, Shawn Cui, Nathan Geer
• arXiv: 2611.06708
• Comments: 34 pages, comments welcome!
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We give invariants of pairs $(M,\Gamma )$ consisting of a closed connected oriented three-manifold and an oriented framed ribbon graph embedded in $M$ colored by a the Drinfeld center of a unimodular chromatic category $\mathcal{C}$. This invariant generalizes the chromatic spherical and renormalized Reshetikhin-Turaev invariants of three-manifolds. We use Heegaard-Graph diagrams which represent the pair $(M,\Gamma )$ and use the data of a unimodular chromatic category and objects of the Drinfeld center to construct the invariant. We define a ribbon graph associated to embedded framed links to obtain an invariant of pairs $(M,L)$. We show that if $L$ is the empty link, then the invariant recovers the chromatic spherical invariant, and if $M$ is the three-sphere, then the invariant recovers the renormalized Reshetikhin-Turaev invariant from the Drinfeld center. The relationship is proven by recovering a chromatic structure for the center induced by that from $\mathcal{C}$. The main theorem of the paper demonstrates that if the graph is a link $L$ colored by a generator in the Drinfeld center, then we recover the chromatic spherical invariant of the surgery manifold $M(L)$, contributing to a proof of the relationship between the chromatic spherical and renormalized Reshetikhin-Turaev invariants for unimodular chromatic categories, generalizing a recent result for unimodular Hopf algebras by Reina. To this end, we give a Heegaard diagram for $M(L)$ coming from the Heegaard-Link diagram representing the pair $(M,L)$.

Cross submissions

Odd and even fake quadrics isogenous to a product

• Authors: Fabrizio Catanese, JongHae Keum, Do Geun Kim, Kyoung-Seog Lee
• arXiv: 2611.06077
• Cross-list: math.AG
• Subjects: Algebraic Geometry (math.AG); Complex Variables (math.CV); Geometric Topology (math.GT)

A $\mathbb{Q}$-homology quadric is a smooth compact complex surface which has the same $\mathbb{Q}$-homology as a quadric surface, that is, the product of two $2$-dimensional spheres. It is called fake if it is minimal and not rational. Among them are the fake quadrics which are isogenous to a product of complex curves. In this paper, we determine the parity of the intersection forms of all fake quadrics isogenous to a product of unmixed type. In particular, we show that there are fake quadrics with odd intersection forms.

Perfect Forms and Torsion Bounds in Algebraic $K$-Theory over Totally Real and CM Fields

• Authors: Alejandro de la Torre Durán
• arXiv: 2611.06300
• Cross-list: math.KT
• Subjects: K-Theory and Homology (math.KT); Geometric Topology (math.GT)

We extend the method of Van Woerden \cite{woerden} to totally real and CM number fields to obtain an upper bound for the number of \GLo-!equivalence classes of perfect forms. Moreover, by considering appropriate barycentric subdivisions of the Voronoi tessellation, we extend these estimates to the number of \GLo!-orbits of $k$-dimensional faces of the Voronoi tessellation. As a consequence, we improve the asymptotic order of the bounds for the torsion part of K_m(\rint)! obtained by Soulé \cite{soule_hermitian_bounds} and Houriet \cite{Juliet_PHD}. We emphasise the improvement of the general bound for , unchanged since Soulé's work, and the independence of the discriminant exponent from the field degree for totally real and CM fields.

A general Lê--Siersma formula

• Authors: R. Giménez Conejero, J.J. Nuño-Ballesteros
• arXiv: 2611.06340
• Cross-list: math.AG
• Comments: Comments very welcome!!
• Subjects: Algebraic Geometry (math.AG); Geometric Topology (math.GT)

We consider the comparison between a Milnor fibre of a function $f$ and its section by the zero locus of a second function $s$ in a general ambient variety $\mathcal{X}$. We give a formula that generalizes those of Lê and Siersma, expressing the difference of their Euler characteristics as a sum of local contributions associated with the critical points of the restriction of $s$. We provide an algebraic formula in the case that $\mathcal{X}$ is {smooth of pure dimension outside }, and show some applications and examples.

Floer theory on based loops and fundamental groups of Lagrangians in

• Authors: Shuhao Li
• arXiv: 2611.06574
• Cross-list: math.SG
• Comments: 21 pages
• Subjects: Symplectic Geometry (math.SG); Algebraic Topology (math.AT); Geometric Topology (math.GT)

We obtain new constraints on the topology of Lagrangian submanifolds in the standard symplectic vector space using Floer theory on the based loop space, generalizing a theorem of Fukaya-Irie. In particular, we show that non-trivial connected sums of closed, spin, aspherical manifolds of even dimensions do not admit Lagrangian embeddings into . A feature of the proof is that the obstructedness of the Floer theory is used to constrain the fundamental group of the Lagrangian.