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Open research questions in Homotopy and Cohomology in Algebraic Topology

66 unresolved questions extracted from the limitations and future-work sections of 337 Homotopy and Cohomology in Algebraic Topology papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • The need to provide a structural reason for preferring one orientation over the other. The challenge of constructing the fundamental representation 3 of SU(3) explicitly from C1–C4.

    Physical Identification of SU(3)_c and the Fundamental Representation 3 from the PDL Axioms C1–C4 (Projective Dynamic Logo Framework — Document D59) · 2026 · DOI

  • The paper identifies a gap in the existing literature on ideal quotients of extriangulated categories. It addresses the need for a relative setting and a new construction of quotients.

    Extriangulated ideal quotients, with applications to cluster theory and gentle algebras · 2026 · DOI

  • The lack of a systematic investigation of coalgebraic models of algebraic theories. The need for a unified framework in which several well-studied algebraic structures fit.

    Coalgebraic Models of $$\Omega $$-Groups · 2026 · DOI

  • The lack of a general construction that produces subcomplexes from any filtered cochain complex of finite depth. The need for a proof that the construction depends only on the given filtration up to isomorphism.

    Filtered complexes and cohomologically equivalent subcomplexes · 2026 · DOI

  • The technical difficulty of proving the result. The need to avoid technical topological assumptions.

    Combinatorial and homotopical aspects of $E_{n}$-operads · 2026 · DOI

  • Further applications of the construction to other areas of mathematics. Study of the cohomology of filtered complexes using spectral sequences.

    Filtered complexes and cohomologically equivalent subcomplexes · 2026 · DOI

  • The lack of a refined theory for justly allocating losses. The inconsistencies in current legal thought between ex ante and ex post perspectives.

    The search for synthesis in tort theory · 1983 · 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

  • The paper identifies a gap in the existing literature regarding the development of a new categorical framework for Galois descent. 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

  • 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 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 need for a more flexible and robust framework than traditional schemes for many modern geometric constructs. The lack of a derived Brauer group for higher stacks.

    Derived Brauer Groups of Higher Stacks and the Homological Obstruction Theory of Categorical Deformations · 2026 · DOI

  • Moreover, while we have focused on second-order obstructions, future research could explore whether higher-order 10 obstructions, typically residing in HHn(C; C ′) for n > 2, can also be related to gen- eralizations of the Brauer group, perhaps via higher algebraic K-theory or other advanced invariants. Further work is needed to precisely delineate these conditions and to explore whether this con- nection holds for arbitrary categorical deformations or if it is restricted to partic- ular classes of A∞-categories or dg-categories. One limitation of the current results is the condition that the deformation be "sufficiently rigid" or possess "specific categorical properties" for the precise iden- tification of the obstruction cocycle with an element of DBr(X ).

    Derived Brauer Groups of Higher Stacks and the Homological Obstruction Theory of Categorical Deformations · 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

  • 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

  • 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 gap in the understanding of noncommutative Poisson geometry and its relationship to quiver varieties. The paper identifies a need for a theory of noncommutative Poisson extensions.

    Shifted double Poisson structures and noncommutative Poisson extensions · 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 paper suggests that further research is needed to fully understand the geometry of algebraic varieties with group actions. The paper suggests that the C-localized Burnside group could be a useful tool for studying the geometry of algebraic varieties with group actions. The paper suggests that the new invariants proposed in the paper could be used to study the problem of determining whether a given action is equivariantly birational to a linear or projectively linear action in higher dimensions.

    Intermediate Jacobians and Burnside invariants · 2026 · DOI

  • Further study of the PDL framework and its applications. Investigation of the implications of the results for quantum chromodynamics.

    Physical Identification of SU(3)_c and the Fundamental Representation 3 from the PDL Axioms C1–C4 (Projective Dynamic Logo Framework — Document D59) · 2026 · DOI

  • To physically identify SU(3)_c. To study the relationship between the three triplets of K4. To derive the fundamental representation 3 from C1-C4.

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

  • The derivation of the algebraic structure SU(3) x SU(2) x U(1) of the Standard Model gauge group was an open problem. The Hubble tension was an open problem. Black hole thermodynamics was not complete.

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

  • The paper suggests that future research should focus on characterizing volume rigidity for all hypergraphs. The paper suggests that future research should explore the applications of the coning lemma for hypergraph volume rigidity.

    Volume Rigidity of Simplicial Manifolds · 2026 · DOI

  • The paper identifies a gap in current research, which is the lack of understanding of volume rigidity for hypergraphs. The paper identifies a need for a coning lemma for hypergraph volume rigidity.

    Volume Rigidity of Simplicial Manifolds · 2026 · DOI

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66 open questions have been extracted from the limitations and future-work passages of 337 Homotopy and Cohomology in Algebraic Topology papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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