Open research questions in Geometric and Algebraic Topology
142 gap statements mined from Geometric and Algebraic Topology papers in our 4.5M-paper local library, which holds 900 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
What the literature leaves open
Further study of the virtual cohomological dimension of subgroups of outer automorphisms is needed. Further study of the relationship between the dimension of the spine and the virtual cohomological dimension is needed.
The relationship between the dimension of the spine and the virtual cohomological dimension was not well understood. The virtual cohomological dimension of the untwisted subgroup U(A) was not well studied.
The existence of an invariant measure for 3D QRT maps was an open question. The paper resolves this question by finding an invariant volume form for 3D QRT maps.
Lastly, in the case of 4-dimensional conformal structures of neutral signature with a conformal Killing field the relation between our Cartan geometric approach and the results of [JT85, DW07] remains to be examined.
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.
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 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.
The conjecture in [Mo2] was previously unresolved. The relationship between h-genus, bridge-1 genus, and Heegaard genus was not fully understood.
The mapping class group of a non-orientable surface is not generated by Dehn twists alone, but also by additional generators such as Y-homeomorphisms and crosscap transpositions. It is not clear if the additional generators can be realized geometrically as monodromies of singular fibers.
Realizing crosscap transpositions as monodromies of singular fibrations · 2026The study of quandles is more difficult than the study of finite groups. There is a lack of methods for computing the quasi-isometry class of a quandle.
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.
Future research could focus on further studying the continuity properties of the boundary and filling functors. Future research could focus on exploring the implications of the results of the paper for the study of rigidity problems in negatively curved spaces.
The model studied here is the combinatorial layer of the Topological Inversion Model, which explores connections between RP3 topology, Skyrmion quantization, and particle physics.
The converse of Makanin's result, whether the existence of infinitely many solutions implies the existence of solutions with arbitrarily large exponent of periodicity, remains open. The problem of identifying structural conditions on groups and their normal forms that guarantee infinite solution sets have an unbounded exponent of periodicity is addressed.
To provide a complete classification of M-fibrations. To explore the implications of M-fibrations for the study of symplectic 4-manifolds and the topology of 4-manifolds.
Realizing crosscap transpositions as monodromies of singular fibrations · 2026To investigate the structure of quandles and their applications.
Further study of the properties of graph products and their applications to quadratic equations in finitely generated groups. Investigation of the implications of the results for the development of algorithms for solving quadratic equations.
The paper identifies a gap in the confirmation of quasi-alternating surgeries on asymmetric L–space knots. Previous results relied on computer-aided findings.
Further study of the properties and applications of the extended knot polynomials. Exploration of the connections between quantum topology and classical topology.
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 rigorous mathematical theory of categorified instanton homology. The application of this theory to the study of exotic smoothness in compact four-manifolds. The exploration of the implications of this theory for other areas of mathematics and physics.
The paper identifies a gap in the understanding of the continuity properties of the boundary and filling functors. The paper identifies a gap in the understanding of the Gromov-Hausdorff convergence of maximal Gromov hyperbolic spaces and their boundaries.
Most-cited papers in Geometric and Algebraic Topology
- Advancing mathematics by guiding human intuition with AI · Nature · 2021 · 402 citations
- The Problem of Tors · Geographical Journal · 1955 · 220 citations
- Acylindrically hyperbolic groups · Transactions of the American Mathematical Society · 2015 · 179 citations
- Hierarchically hyperbolic spaces, I: Curve complexes for cubical groups · Geometry & Topology · 2017 · 82 citations
- An Elementary Introduction to the Hopf Fibration · Mathematics Magazine · 2003 · 60 citations
- Torsion of the Fallopian Tube · Acta Obstetricia Et Gynecologica Scandinavica · 1962 · 55 citations
- A Tiebout theorem · Mathematical Social Sciences · 1989 · 53 citations
- The Staple Theory Revisited · Journal of Canadian Studies · 1977 · 44 citations
- A characterization of the single-crossing domain · Social Choice and Welfare · 2012 · 38 citations
- The New Polynomial Invariants of Knots and Links · Mathematics Magazine · 1988 · 31 citations
Most recent work
- Relating Flat Connections and Polylogarithms on Higher Genus Riemann Surfaces · Communications in Mathematical Physics · 2026
- Instability of Legendrian knottedness, and non-regular Lagrangian concordances of knots · Advances in Mathematics · 2026
- Slope detection and toroidal 3-manifolds · Advances in Mathematics · 2026
- Self-simulable groups · Transactions of the American Mathematical Society · 2026
- On constant mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds. · Calculus of Variations and Partial Differential Equations · 2026
- Regularity of Resolutions and Limits of Manifolds with a Uniform Contractibility Function · The Journal of Geometric Analysis · 2026
- Satellite knots and immersed Heegaard Floer homology · Geometry & Topology · 2026
- An orbifold framework for classifying layer groups with an application to knitted fabrics · Interface Focus · 2026
- High symmetry embeddings of rectilinear tangles · Interface Focus · 2026
- On Multiplicities in Length Spectra of Semi-Arithmetic Hyperbolic Surfaces · Communications in Mathematical Physics · 2026
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