Studies amenable category's monotonicity and its relation to topological complexity.
problem Monotonicity of amenable category for degree-one maps.
method Uses amenable covers and compares with topological complexity.
result Establishes a relation between amenable category and topological complexity.
Formulates a new connection between topological and geometric categories.
problem No specific problem stated; focuses on category formulation.
method Formulates a connection between a topological and geometric category.
result Provides a more precise and improved version of a previous proposal.
The study explores how different Grothendieck topologies and functors between categories preserve locality.
problem Exploring relationships between different Grothendieck topologies and functors.
method Using Grothendieck topologies and functors to relate categories and geometric objects.
result Objects like sheaves, groupoids, and functors are invariant under equivalences of Grothendieck topologies and certain functors.
ETQFTs created from non-semisimple modular categories.
problem Constructing ETQFTs from non-semisimple modular categories.
method Explicitly identify linear categories and functors in the image of ETQFTs constructed from modular categories.
result The circle category of ETQFTs is equivalent to the full subcategory of projective objects of the underlying modular category, which need not be semisimple.
This is a survey on two closely related subjects. First, we review the study of topological structure of `finite type' components of spaces of Bridgeland's stability conditions on triangulated categories. The key is to understand Happel-Reiten-Smalo tilting as tiling of cells. Second, we review topological realizations…
The paper explores how the unit inclusion affects topological quantum field theories in non-semisimple categories.
problem Understanding the effects of unit inclusion in non-semisimple braided tensor categories on topological quantum field theories.
method Analyzes the dualizability of the unit inclusion morphism in Morita 4-category of braided tensor categories and applies the Cobordism Hypothesis.
result Shows that the unit inclusion in non-semisimple modular categories leads to non-compact relative 3D topological quantum field theories.
Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the categ…
Recent work extends Turaev's modular categories to non-semisimple settings.
problem Building TQFTs beyond semisimplicity.
method Generalized modular categories.
result Success in extending Turaev's construction to non-semisimple settings.
We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity TC(X) and monoidal topological complexity TCM(X). Using these results we provide lower and upper bounds for the topological complexity of the wedge X∨Y. We use these bounds to give a counterexample t…
Constructs dg categories from surfaces using Khovanov homology.
problem Categorify quantum topology using surfaces and Khovanov homology.
method Constructs dg categories from surfaces using structures in Khovanov homology.
result Unified perspective on various categorified quantum topology constructions.
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-di…
New probabilistic invariants bound classical topological complexity and category.
problem Bounding classical topological complexity and category.
method Developed probabilistic variants of one-category and diagonal topological complexity.
result Identified new invariants with distributional category and complexity on Eilenberg-Mac Lane spaces.
Survey of topological 4-manifold theory, highlighting foundational theorems and pitfalls.
problem Understanding the topological properties of 4-dimensional manifolds.
method Compilation and explanation of foundational theorems, with cautionary notes.
result Many intuitive results in differential topology are not true in the topological category.
Topological complexity for closed 1-forms
problem Topological complexity for closed 1-forms
method Introduce and study a corresponding version of topological complexity
result Establish analogues of basic properties of ordinary topological complexity
Study proves topological complexity and LS-category inequalities for specific groups and manifolds.
problem Proving inequalities for topological complexity and LS-category of specific groups and manifolds.
method Analyzing torsion free hyperbolic and nilpotent groups, lens spaces, using inequalities and counter-examples.
result Proves inequalities for topological complexity and LS-category of specific groups and manifolds.
The article applies Lusternik-Schnirelmann theory to establish lower bounds on critical points using sequential and parametrized topological complexity.
problem Establishing lower bounds on the number of critical points of functions using topological complexity.
method Applying Lusternik-Schnirelmann theory to sequential and parametrized topological complexity.
result Established various lower bounds on the number of critical points using sequential and parametrized topological complexity.
In this paper we study the topology of the cobordism category of open and closed strings. This is a 2-category in which the objects are compact one-manifolds whose boundary components are labeled by an indexing set (the set of "D-branes"), the 1-morphisms are cobordisms of manifolds with boundary, and the 2-morphisms a…
Study of universal complexes in toric topology with applications in category theory.
problem Properties and applications of universal complexes in toric topology.
method Combinatorial and topological analysis of X(Fpn) and K(Fpn). result Lusternick-Schnirelmann categories of moment angle complexes calculated for universal complexes.
We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.
New TC variant dTC better fits motion planning for some systems.
problem Improving motion planning for autonomous systems.
method Defined and computed new homotopy invariant dTC.
result dTC and dcat provide better motion planning solutions.
This paper describes how to recover the topology of a closed manifold M from a good Morse function f on M. The essential method was suggested by Cohen, Jones and Segal. They constructed a topological category Cf and claimed that the classifying space BCf is homeomorphic to M. We prove it from a differ…
Study probabilistic category and complexity bounds, comparing with classical invariants.
problem Bounding classical category and complexity in probabilistic settings.
method Probabilistic Lusternik-Schnirelmann category and topological complexity computations.
result Established a universal upper bound in finite cases, contrasting with classical invariants.
Extends Gelfand duality to various geometric and analytical categories.
problem Generalizing Gelfand duality to different types of manifolds and bundles.
method Unified cohomological argument for manifolds and suitable classes of functions for bundles.
result Gelfand duality extended to real analytic and Stein manifolds, and to vector, affine, and jet bundles.
New coarse LS-category introduced for groups and spaces.
problem Large-scale topological properties of groups and spaces.
method Introducing a coarse analog of Lusternik-Schnirelmann category for metric spaces.
result Established lower and upper bounds for geometrically finite and bicombable groups.
Mathematical study supports connection between 3D manifolds and modular tensor categories.
problem Connecting geometric topology and quantum topology using Chern-Simons invariants and Reidemeister torsions.
method Developed an algorithm to generate modular T-matrices and quantum dimensions from Seifert fibered spaces and torus bundles over the circle. result Mathematically constructed premodular categories from Seifert fibered spaces and torus bundles over the circle, conjecturing their modularity under specific conditions.
The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\c…
Study numerical invariants under retraction maps between topological spaces.
problem Understand behavior of invariants like cohomological dimensions under retractions.
method Introduced a notion of retraction and studied several numerical invariants.
result Proved inequalities between invariants hold under retractions.
A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible simple objects. This paper gives a detailed exploration of "topological quantum …
In this paper, we generalize the notion of Serre fibration to the Morita category of topological groupoids and derive the associated long exact sequence of homotopy groups. We use this results for calculation of homotopy groups of various groupoids, such as the foliation groupoid of a Riemannian foliation.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
Study on topological rigidity of ALE vector bundles with specific conditions.
problem Classifying ALE vector bundles with asymptotically conical total spaces.
method Topological classification and geometric analysis of ALE vector bundles.
result Only 2-sphere, projective plane, and open contractible manifolds admit ALE tangent bundles.
We define a cobordism category of topological manifolds and prove that if d=4 its classifying space is weakly equivalent to Ω∞−1MTTop(d), where MTTop(d) is the Thom spectrum of the inverse of the canonical bundle over BTop(d). We also give versions with tangential structures and boundary. The pro…
We consider the topological category of h-cobordisms between manifolds with boundary and compare its homotopy type with the standard h-cobordism space of a compact smooth manifold.
Study homotopy sheaves on categories and their presheaves, proving descent properties.
problem Homotopy sheaves on categories and their presheaves.
method Homotopy right Kan extension, pretopologies, Yoneda embedding.
result Preserves homotopy sheaves and induces equivalence between sheaves and colimit-preserving sheaves.
We introduce the notion of a relative spherical category. We prove that such a category gives rise to the generalized Kashaev and Turaev-Viro-type 3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624, respectively. In this case we show that these invariants are equal and extend to what we call a relativ…
This paper introduces quantum invariants for 3-alterfolds and proves their consistency with topological moves.
problem Quantum invariants for 3-alterfolds and their consistency with topological moves.
method Introduction of 3-alterfolds with embedded separating surfaces and spherical fusion categories.
result Quantum invariants of 3-alterfolds are consistent with topological moves and generalize invariants of 3-manifolds containing framed links.
We formalize the arithmetic topology, i.e. a relationship between knots and primes. Namely, using the notion of a cluster C*-algebra we construct a functor from the category of 3-dimensional manifolds M to a category of algebraic number fields K, such that the prime ideals (ideals, resp.) in the ring of integers of K c…
We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the (1+1)-dimensional topological quantum field theory defined by Honda--Kazez--Matić in \cite{HKM08}. The Z2 SFH of these solid tori forms a "categorification of Pascal's triangle", and contact structur…
The paper extends topological field theory to noncompact surfaces using symmetric powers.
problem Extending topological field theory to noncompact surfaces without closed boundaries.
method Constructing sectorial covers with combinatorics of the bar resolution.
result Recovering results of Rouquier and Manion on extending Heegaard-Floer theory.
Develops persistent Khovanov homology for tangles.
problem Lack of local topological features in evolutionary Khovanov homology for knots and links.
method Introduces a new mathematical framework using persistent Khovanov homology of tangles, employing functor and planar algebra.
result Provides a new method to characterize local features in curve-type data.
Homotopy Quantum Field Theories (HQFTs) generalize more familiar Topological Quantum Field Theories (TQFTs). In generalization of the surgery construction of 3-dimensional TQFTs from modular categories, we use surgery to derive 3-dimensional HQFTs from G-modular categories.
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations…
Defines Whitehead torsion for topological spaces via K-theory.
problem Classical Whitehead torsion for topological spaces.
method Introduces Whitehead categories and torsion cosheaves.
result Establishes a connection between Whitehead torsion and K-theory.
A new method analyzes topological B-model on a torus using doubled geometry.
problem Reproduce derived category of coherent sheaves on a torus.
method Double field theory and T-duality in doubled geometry framework.
result Correctly computes BRST cohomology and derived Hom-spaces of line bundles.
The LS-category of a topological space is a numerical homotopy invariant, introduced originally in a course on the global calculus of variations by Lyusternik and Schnirelmann, to estimate the number of critical points of a smooth function. When the topological space is a smooth manifold equipped with a proper action o…
The study finds conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
problem Conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
method Analyzes conditions and examples to generalize results on topological complexity and Lusternik-Schnirelmann category.
result Generalizes results on topological complexity and Lusternik-Schnirelmann category for manifolds with abelian fundamental groups.
Unified approach to homological representations of topological groups.
problem Constructing homological representations of topological groups.
method Functorial approach using topological enrichment of Quillen bracket construction.
result Unified construction of homological representations for mapping class groups and motion groups.
The paper explores the topology and curvature of isoparametric families in spheres.
problem Investigating the topology and curvature of isoparametric families in spheres.
method The paper investigates the topology and curvature of isoparametric families in spheres using homotopy, homeomorphism, diffeomorphism types, parallelizability, and Lusternik-Schnirelmann category.
result The paper determines conditions for non-negative sectional curvatures and positive Ricci curvatures in isoparametric families.