Open research questions in Homotopy and Cohomology in Algebraic Topology

What the literature leaves open

• The lack of a systematic way to study cohomology pre-algebraic structures via classical cohomology. The need to establish relationships between Perm algebras, pre-associative algebras, and pre-Lie algebras.

  A New Approach to Defining Cochain Complexes for Dendriform and Pre-Lie Algebras · 2026 · DOI

• The need to understand the properties of B-polytopes and their relationship to V-faces and B-faces.

  B-facets in Dimension 4 · 2026 · DOI

• The paper aims to address this gap by establishing a Galois descent theorem and providing a sufficient condition for a functor to be a categorical equivalence.

  Galois descent theorem in general categories · 2026 · DOI

• The gap in existing approaches is the requirement of finiteness constraints. The paper identifies the need for a framework that extends homotopical invariants to small categories without these constraints.

  Weak and Strong Fibrations of Functors · 2026

• Future research could involve applying the results of the paper to other areas of algebraic geometry. Future research could involve studying the properties of the semicosimplicial dgLa constructed in the paper.

  Deformations of morphisms of coherent sheaves · 2026

• The paper identifies a need for a new tool for studying the infinitesimal deformations of a morphism of coherent sheaves.

  Deformations of morphisms of coherent sheaves · 2026

• The need for a unified framework for Bloch's theorem, vortex circulation, and electrodynamics. The lack of a deep understanding of the interplay between characters, residues, and analytic mappings.

  Algebraic and Topological Approaches to Bloch's Theorem, Vortex Circulation, and Electrodynamics via Complex Analysis · 2026 · DOI

• The lack of examples of phantom categories on rational surfaces with full exceptional collections. The need to explore the properties of phantom categories, including their height and pseudoheight.

  A looming of phantoms · 2026 · DOI

• The framework is limited to planar graphs and does not provide a general solution to the problem. The existence-consistency mismatch is a fundamental obstacle in cohomological approaches to the Four-Color Theorem.

  An Explanatory Framework for the Four-Color Problem via F₂²-Flows, Local Conservation Laws, and the Existence–Consistency Mismatch · 2026 · DOI

• Future research should focus on developing new algorithms and techniques for solving the Four-Color Problem using the proposed framework. Future research should also focus on identifying new areas of research and developing new applications for the problem.

  An Explanatory Framework for the Four-Color Problem via F₂²-Flows, Local Conservation Laws, and the Existence–Consistency Mismatch · 2026 · DOI

• Future research could explore the applications of the theory of noncommutative Poisson extensions. Future research could investigate the relationship between noncommutative Poisson geometry and other areas of mathematics.

  Shifted double Poisson structures and noncommutative Poisson extensions · 2026 · DOI

• The paper identifies a need for a theory of noncommutative Poisson extensions.

  Shifted double Poisson structures and noncommutative Poisson extensions · 2026 · DOI

• The lack of a general method for constructing finite semigroups with specific classifying spaces. The inability to realize wedges of Moore spaces as classifying spaces of finite semigroups, particularly in the context of simply connected CW complexes. The need for a deeper understanding of the relationship between finite semigroups and CW complexes.

  Realizing wedges of Moore spaces as classifying spaces of finite semigroups · 2026 · DOI

• The paper does not provide a general formula for the computation of higher limits over orbit categories of locally finite groups. The results are limited to the case of countable locally finite groups with finite Sylow p-subgroups.

  Limits over orbit categories of locally finite groups · 2026 · DOI

• The paper suggests that future research could focus on the computation of higher limits over orbit categories of more general classes of groups. The authors also suggest that the results of the paper could be applied to a variety of contexts in algebra and geometry.

  Limits over orbit categories of locally finite groups · 2026 · DOI

• The lack of a unified framework for classifying thick submodules in stable infinity categories with actions. The need for a new invariant that generalizes existing spectra and provides a unified framework for classifying thick submodules.

  The Spectrum of Stable Infinity Categories with Actions · 2026 · DOI

• To explore the existence of phantom subcategories in other rational surfaces. To study the properties of phantom categories in greater detail. To construct more examples of phantom categories in derived categories of varieties.

  A looming of phantoms · 2026 · DOI

• The paper does not provide a complete classification of actions on smooth projective rational varieties. The paper focuses on threefolds and does not address the problem in higher dimensions. The paper does not provide a explicit algorithm for computing the class of the G-action.

  Intermediate Jacobians and Burnside invariants · 2026 · DOI

• The Indifference Lemma (H3) has been derived from axioms C1–C4 via three lemmas: D42-L1 (Equiparticipation), D42- L2 (Cross-Triangle Blindness), D42-L3 (S4-Equivariance). As a corollary, κ = 310φ/11017 ∈ Q( 5) is an unconditional theorem. The entire causal chain from C1–C4 to the Bekenstein–Hawking entropy is now fully axiomatic.

  Projective Dynamic Logo (PDL) — Global Mapping of Structures, Results, and Open Problems (Version 26) · 2026 · DOI

• Further study of the cohomology of Perm algebras and pre-Lie algebras. Exploration of the connections between pre-associative algebras and operad theory.

  A New Approach to Defining Cochain Complexes for Dendriform and Pre-Lie Algebras · 2026 · DOI

• The paper identifies a gap in the literature on V-graded categories and bigraded squares, - It seeks to develop general results on bigraded squares in V-W-bigraded…

  V-graded categories and V-W-bigraded categories: Functor categories and bifunctors over non-symmetric bases · 2026 · DOI

• To generalize the classification of B-facets to higher dimensions. To study the applications of the classification of B-facets in algebraic geometry and singularity theory.

  B-facets in Dimension 4 · 2026 · DOI

• Further study of the applications of the paper's results to algebraic geometry and category theory. Exploration of the connections between the paper's results and other areas of mathematics.

  Galois descent theorem in general categories · 2026 · DOI

• Extensions to nonlinear field theories. Applications to modern celestial-CFT approaches to scattering amplitudes.

  Algebraic and Topological Approaches to Bloch's Theorem, Vortex Circulation, and Electrodynamics via Complex Analysis · 2026 · DOI

• The paper identifies a gap in the study of Symmetry Topological Field Theory (SymTFT) for continuous 0-form G-symmetry. The paper also identifies a gap in the study of the Drinfeld center and its applications.

  Categorical continuous symmetry · 2026 · DOI

Most-cited papers in Homotopy and Cohomology in Algebraic Topology

• A purely geometric module in the rat's spatial representation · Cognition · 1986 · 791 citations
• The Assignment of Numbers to Rank Order Categories · American Sociological Review · 1970 · 419 citations
• Elementary Linkage Analysis for Isolating Orthogonal and Oblique Types and Typal Relevancies · Educational and Psychological Measurement · 1957 · 276 citations
• Hylomorphism · The Journal of Philosophy · 2006 · 154 citations
• Factors involved in linking representations of functions · The Journal of Mathematical Behavior · 1998 · 148 citations
• Partiality and Its Dual · Studia Logica · 2000 · 100 citations
• Conjectural variations as a reduced form · Economics Letters · 1995 · 67 citations
• Arguments for adjuncts · Cognition · 2003 · 55 citations
• The sheaf coefficient: A simplified and expanded approach · Social Science Research · 1986 · 51 citations
• Diastrophism as the Ultimate Basis of Correlation · The Journal of Geology · 1909 · 51 citations

Most recent work

• Physical Identification of SU(3)_c and the Fundamental Representation 3 from the PDL Axioms C1–C4 (Projective Dynamic Logo Framework — Document D59) · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Projective Dynamic Logo (PDL) — Global Mapping of Structures, Results, and Open Problems (Version 26) · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Carrollian Lie algebroids: taming singular Carrollian geometries · Classical and Quantum Gravity · 2026
• The de Rham cohomology of a Lie group modulo a dense subgroup · Proceedings of the American Mathematical Society · 2026
• The 3d Mixed BF Lagrangian 1-Form: A Variational Formulation of Hitchin’s Integrable System · Communications in Mathematical Physics · 2026
• Towards the dual motivic Steenrod algebra in positive characteristic · Mathematische Zeitschrift · 2026
• Symplectic Morse theory and Witten deformation · Mathematische Zeitschrift · 2026
• Alterfold Theory and Topological Modular Invariance · Communications in Mathematical Physics · 2026
• Analog of Satake–Baily–Borel for period maps · European Journal of Mathematics · 2026
• Positive and negative extensions in extriangulated categories · Mathematische Zeitschrift · 2026

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