Open research questions in Geometric and Algebraic Topology
75 unresolved questions extracted from the limitations and future-work sections of 370 Geometric and Algebraic Topology papers in our library. Each links back to the study that raised it.
What the literature leaves open
The paper suggests that future research should focus on developing an algorithmic approach to studying translation length functions arising from free actions on R-trees. The paper suggests that future research should focus on establishing a connection between pseudo-lengths and discrete free groups of isometries.
The paper identifies a gap in the theory of translation length functions for two-generated groups acting on trees. The paper identifies a need for an explicit formula for the translation length of any element of a two-generated group acting on a tree.
The paper does not provide a general solution to the extension problem for all GKM manifolds. The proof of Theorem 4.19 is limited to (n, k)-type GKM3 graphs with countably many vertices and edges.
To study the extension problem for GKM manifolds of higher complexity. To generalize the results to other types of GKM manifolds.
The development of a categorified instanton homology is a formidable challenge. Sophisticated machinery from higher category theory and novel analytical techniques for infinite-dimensional moduli spaces are required. The smooth classification of compact four-manifolds remains notoriously difficult and incomplete.
The paper identifies a gap in the understanding of the primeness of alternating virtual links. The paper identifies a gap in the understanding of the correspondence between cellular link diagrams on closed surfaces and equivalence classes of virtual link diagrams.
Prior to this study, the residual properties of non-trivial knot groups were not fully understood. The relationship between hyperbolic knots and closed hyperbolic 3-manifold groups needed further exploration.
The classification of exact Lagrangian fillings of Legendrian links is an important problem in low-dimensional contact and symplectic topology. There is a lack of understanding of constructing and distinguishing fillings of Legendrian links.
Future research could focus on providing a complete mathematical formulation of the theory. Future research could also focus on studying the potential applications of the theory in cosmology and quantum gravity. Future research could also focus on numerical simulations of the theory.
The gap is the lack of a coherent theory of the chronogeometric evolution of universes. The gap is also the lack of a complete mathematical formulation of the theory.
There are only two known strongly invertible L-space knots with a certain property. The paper aims to fill this gap by finding another example.
The paper identifies a gap in understanding how two-focus curves unify dual images.
Further study of the residual properties of knot groups. Exploration of the applications of the results in topology and other fields.
Applying the unified framework to other complex dynamical systems. Exploring potential applications to other areas of physics.
The Hyperbolic, Eisenstein–Lattice, and FRLT Interpretation of the Figure‑Eight Choreography in the Three‑Body Problem · 2026 · DOIThe lack of a unified framework for complex dynamical systems. The lack of a geometric framework for the three-body problem.
The Hyperbolic, Eisenstein–Lattice, and FRLT Interpretation of the Figure‑Eight Choreography in the Three‑Body Problem · 2026 · DOI5 Connections and open problems The model studied here is the combinatorial layer of the Topological Inversion Model [12], which explores connections between RP3 topology, Skyrmion quantization, and particle physics.
The conditions for reducing holonomy to proper subgroups of the contact parabolic subgroup P ⊂ B_3 are specified algebraically (vanishing of b_1, b_2, b_3, b_5, b_6) in part (3) of Corollary 5.11, but the geometric meaning and practical implications of such holonomy reduction for the underlying (3,6)-distributions with null infinitesimal symmetry are not discussed.
The paper derives fundamental invariants (b_i coefficients) for pairs of third order ODEs through explicit parametric expressions (equations 5.59-5.60) and uses Cartan-Kähler analysis to determine local generality, but does not provide explicit examples beyond the degenerate Lagrangian case demonstrating how to compute or verify these invariants for concrete ODE systems.
The admissible normal form nf: G → Γ* is assumed to exist for finitely generated torsion-free groups with the finite square roots property, but the paper does not provide explicit constructions or algorithms for computing such normal forms for specific graph product classes (e.g., right-angled Artin groups, free products), limiting practical application of Theorem 6.1.
Case (4) of Theorem 6.1 shows that when a variable occurs only once in a quadratic equation W, the solution set contains the product ⋃{LY | Y ∈ X ∖ {X, X⁻¹}} with infinite exponent of periodicity, but the paper does not explore whether tighter bounds on exp(nf(Sol(S,µ))) can be achieved by analyzing the specific structure of the remaining constraint variables or their interaction patterns.
The paper identifies a gap in the literature regarding the construction of hyperbolic surfaces with large systole and kissing number. The paper also identifies a gap in the literature regarding the generalization of Theorem 2.6 to non-compact hyperbolic manifolds of finite volume.
The paper notes that the models may learn to classify knots based on features that are not invariant under ambient isotopy. The paper notes that the dataset is limited to an ensemble of unknots and trefoil knots. The paper notes that the models are trained on a simplified classification challenge.
The paper suggests that future research should focus on developing machine learning models that can solve complex geometric knot classification challenges. The paper suggests that future research should investigate the use of more complex machine learning models and larger datasets. The paper suggests that future research should explore the application of machine learning to other areas of low-dimensional topology.
The lack of a description of Anosov diffeomorphisms that are smoothly conjugate to algebraic models in higher dimensions. The higher dimensional case is more complicated and answers to questions of regularity of conjugacy are negative in general without some irreducibility assumption on L.
Further study of the properties and applications of the extended knot polynomials. Exploration of the connections between quantum topology and classical topology.
Most-cited papers in Geometric and Algebraic Topology
- The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle · Topological Methods in Nonlinear Analysis · 2020 · 2 citations
- On topologically distinct infinite families of exact Lagrangian fillings · Archivum Mathematicum · 2022 · 1 citations
- The Borsuk-Ulam property for homotopy classes of maps from the torus to the Klein bottle - part 2 · Topological Methods in Nonlinear Analysis · 2022 · 1 citations
- The Borsuk-Ulam property for maps from the product of two surfaces into a surface · Topological Methods in Nonlinear Analysis · 2021 · 1 citations
- On Reeb graphs induced from smooth functions on 3-dimensional closed manifolds with finitely many singular values · Topological Methods in Nonlinear Analysis · 2022 · 1 citations
- Canonical Classication of Framed Closures in a Three-Strand Braid Transfer Model · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 0 citations
- Realising VCD for untwisted automorphism groups of RAAGs · Geometriae Dedicata · 2026 · 0 citations
- Invariant Volume Form for 3D QRT Maps · Mathematical Physics, Analysis and Geometry · 2026 · 0 citations
- Parabolic Quasi-Contact Cone Structures with an Infinitesimal Symmetry · Transformation Groups · 2026 · 0 citations
- Quadratic Equations over Graph Products and the Exponent of Periodicity · journal of Groups complexity cryptology · 2026 · 0 citations
Most recent work
- Canonical Classication of Framed Closures in a Three-Strand Braid Transfer Model · Zenodo (CERN European Organization for Nuclear Research) · 2026
- Realising VCD for untwisted automorphism groups of RAAGs · Geometriae Dedicata · 2026
- Invariant Volume Form for 3D QRT Maps · Mathematical Physics, Analysis and Geometry · 2026
- Parabolic Quasi-Contact Cone Structures with an Infinitesimal Symmetry · Transformation Groups · 2026
- Quadratic Equations over Graph Products and the Exponent of Periodicity · journal of Groups complexity cryptology · 2026
- Extension of Wada’s Representation of the Pure Braid Group on n Strings · Vietnam Journal of Mathematics · 2026
- On Systole, Kissing Number and Volume of Arithmetic Manifolds · Communications in Mathematics · 2026
- Shortcut learning in geometric knot classification · Machine Learning: Science and Technology · 2026
- Global smooth rigidity for toral automorphisms · Inventiones mathematicae · 2026
- On the conjugacy problem for subdirect products of hyperbolic groups · Mathematische Annalen · 2026
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