math.GT daily digest: 8 submissions for 16 September 2026

New submissions

Incidence complexes realization and profinite Rigidity

• Authors: Yong Hou
• arXiv: 2609.16020
• Comments: 56 pages
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

We prove realization theorems for marked profinite groups by means of divisible fillings and completed cellular-incidence complexes. For a nonseparating curve on a closed surface, the divisible curve-power quotients realize the closure of the cut-surface subgroup as an exact intersection and determine the full profinite cut tree. For a nonseparating filling pair, the completed dual square complex is the closed crossing subcomplex of the product of the two cut trees. Its incidence maps realize the given discrete surface action up to a single profinite translation. The same argument applies to free cocompact actions on connected, locally finite, finite-dimensional regular CW complexes. A graph version permits finite vertex stabilizers and assumes equivariant incidence isomorphisms only at cofinally many characteristic finite quotients. The finite-intersection property supplies compatible maps, and the same construction identifies the outer automorphism groups of the discrete and completed marked actions. Full simplex-face incidence yields PL realization of finite simplicial complexes. We apply these results to Kleinian groups. In the lattice case, an integral cohomological comparison, Massey triple products, and an exact correspondence of prime periodic orbits produce the required surface markings, while unit-invariant divisible orbifold fillings treat the cusped case. For arbitrary finitely generated Kleinian groups, equivariant graph markings give boundary-subgroup correspondence and PL realization of marked compact cores. A Schottky example shows that such markings cannot in general be omitted outside the lattice setting.

The underlying manifold of a low-volume hyperbolic 3-orbifold

• Authors: David Futer; Peter B. Shalen
• arXiv: 2609.16212
• Comments: 100 pages, 2 figures
• Subjects: Geometric Topology (math.GT)

This paper proves strong restrictions on the underlying topological space of a closed, orientable hyperbolic 3-orbifold \orbM whose volume satisfies a certain upper bound. For instance, if vol(\orbM) < 0.1491, then the underlying space of \orbM must be either a small Seifert fibered space, or the connected sum of two lens spaces (or of a lens space with S^2 x S^1), or the gluing of one or two highly restricted Seifert fibered spaces along a single incompressible torus. If vol(\orbM) < 0.1571 and the singular locus of \orbM is a link, then the topology of the underlying space is restricted even further.

Our methods are primarily topological, and involve studying the underlying topology of orbifold books of I-bundles. Along the way, we provide an exposition of some foundational material about orbifolds that was previously absent from the literature.

Splice formulae for Poincaré series of integral homology spheres

• Authors: Tamás László; András Némethi; György Tőtős
• arXiv: 2609.16840
• Comments: 30 pages
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG)

Let M be a plumbed integral homology sphere 3-manifold associated with a connected negative definite plumbing graph $\Gamma$. One of its most important invariants is its multivariable Poincaré series (or zeta function) $f\_\Gamma (\mathbf{t})$. Several numerical invariants can be read from $f\_\Gamma (\mathbf{t})$, or even from its `polynomial part' $ \operatorname{Pol}_{\Gamma}(\mathbf{t})$. For example, the `normalized' Casson's invariant equals $ \operatorname{Pol}_{\Gamma}(1)$.

In this note we provide several splice (surgery) formulae for $f\_\Gamma$ and $\mathrm{Pol}\_\Gamma$ (reduced to the node variables, or to the node variables of connected sub-graphs). In this way, these global invariants can be recovered from a collection of certain smaller graphs.

Recall that some (integral homology sphere) 3-manifold invariants are additive with respect to the splice decomposition (like the Casson's invariant). However, some invariants need some `splice correction terms'. In our formulae the correction terms are easily computable one variable Alexander polynomials.

Deciding strong quasipositivity and quasipositivity

• Authors: Marc Kegel; Qiuyu Ren
• arXiv: 2609.17341
• Comments: 7 pages
• Subjects: Geometric Topology (math.GT)

We present two results concerning the decidability of (strong) quasipositivity of links in the 3-sphere. First, using a result of Dynnikov--Prasolov, we provide an algorithm to decide whether or not a link is strongly quasipositive. Second, we reduce the decision problem of the quasipositivity of links to that of braids.

Computations of Floer Lasagna Modules for Traces of Negative L-Space Knots

• Authors: Colin McCulloch
• arXiv: 2609.17503
• Comments: 47 pages, 11 figures, 2 tables. Comments welcome!
• Subjects: Geometric Topology (math.GT)

We show that for all negative $L$-space knots and framings , if a certain cobordism map between cables of $K$ is non-vanishing, then the Floer lasagna module of the knot trace of $K$ with framing $-n$ is infinite dimensional. Further, for the maps on the hat flavour of link Floer homology induced by band maps, quasi-stabilisations and pair of pants cobordisms we give a description of the map on the -page of Ozsvath and Szabo's spectral sequence for forgetting components.

Cross submissions

On smooth and bundle structures on topological manifolds homotopy equivalent to -bundles over

• Authors: Tibor Macko; Ajay Raj
• arXiv: 2609.16307
• Cross-list: math.AT
• Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)

We first verify the expected calculations of the topological structure set, in the sense of surgery theory, for the total spaces of -bundles over for , in which manifolds homotopy equivalent to the total spaces are organized. Next, we investigate the question of which of the elements in these structure sets can be realized as such bundles. Moreover, we study the forgetful map from the smooth version to the topological version of the structure set; we determine its image. All elements in the image turn out to have the underlying manifold such a bundle.

Milnor fibrations on rho-tubes and rho-spheres for real analytic map germs

• Authors: Maico Ribeiro; Mateus de Melo; Gabriel Santos
• arXiv: 2609.17374
• Cross-list: math.DG
• Comments: 17 pages,
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

For a real analytic map germ , we study Milnor fibrations obtained by replacing the squared Euclidean distance with an analytic control function $\rho$ defining the origin. We formulate the corresponding Milnor set and condition (b), derive criteria for tube and sphere fibrations, and examine their dependence on $\rho$. The family provides explicit changes of regularity under elliptic controls. In particular, the germ admits neither the induced tube nor sphere fibration for the Euclidean control, while both fibrations exist for the adapted function . Thus the control function can be essential to the local fibration structure.

Quandles associated with group actions

• Authors: Ryoya Kai
• arXiv: 2609.17438
• Cross-list: math.GR
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

A quandle is an algebraic system that can be regarded as a generalization of the conjugation operation in groups. In this paper, we establish a Cayley-type embedding theorem for finite quandles. To this end, we study a quandle construction associated with group actions and determine its fundamental structural properties. Applying the construction to the natural action of the symmetric group, we obtain a quandle into which every quandle of cardinality $n$ embeds.