67 papers
This paper introduces discrete approximate circle bundles as a data science analog to topological circle bundles, providing stable identification methods, cohomology invariants, and computational algorithms for coordinatization and dimensionality reduction, which are demonstrated through applications in computer vision and supported by an open-source software package.
This paper establishes that the -category of normed algebras in genuine -spectra is modeled by strictly commutative algebras in -symmetric spectra, providing a higher-categorical description of ultra-commutative global ring spectra as a partially lax limit of genuine -spectra categories.
This paper extends fundamental homotopical concepts to -categories and presentable enriched categories by introducing homotopy posets indexed by categorical disk boundaries, which assemble into a Postnikov tower converging for -categories and characterize Postnikov complete -categories as the limit of -categories under truncation.
This paper constructs a fractured structure on the -topos of condensed anima to identify a jointly conservative collection of points and resolves a question by Clausen by demonstrating that the category of extremally disconnected spaces lacks all fibers.
This paper determines the homotopy type of the moment-angle complex associated with the complex of injective words by showing it is governed by the complex's -vector and establishes a generalized homotopy fibration for polyhedral products over ordered simplicial complexes.
This paper extends the cohomological and spectral sequence methods for estimating the topological complexity of non-simply connected spaces via group homomorphisms, applying these results to determine the topological complexity of specific 3-manifolds with nonabelian fundamental groups.
This paper establishes that Szczarba's twisted shuffle restricts to a chain isomorphism between path chain complexes in the marked setting, thereby enabling the definition and explicit computation of equivariant path homology for directed graphs with group actions via a natural Borel construction.
This paper reformulates the Chern character in topological K-theory using Fermi points of Fredholm operators to generalize spectral flow and provide elementary proofs for the evenness of the edge index and the bulk-edge correspondence in four-dimensional time-reversal symmetric topological insulators (class AI).
This paper formalizes the Galois correspondence between covering spaces and subgroups of fundamental groups within homotopy type theory, extends the concept to n-dimensional generalizations, and applies the framework to classify lens space coverings and construct the Poincaré homology sphere.
This paper investigates homotopy decompositions of classifying spaces of compact connected Lie groups via a relative fiber-cofiber construction, establishing rational sharpness and Cohen-Macaulay properties for the resulting spaces under specific cohomological conditions while providing explicit cohomological and K-theoretic computations for examples involving maximal tori and commuting elements.
This paper demonstrates that a canonical extension of generalized Dynkin diagrams generates families of Kac-Moody groups, such as the series relevant to string theory, which satisfy homological stability and exhibit emergent structures upon stabilization, utilizing homotopy decompositions of their classifying spaces.
This paper resolves the Peterson hit problem for five variables in generic degrees to prove that the fifth Singer algebraic transfer is an isomorphism in an infinite family of degrees, validates a localized version of Kameko's conjecture, and distinguishes the homotopy types of specific complex projective space quotients via their Steenrod module structures.
This paper establishes that motivic stable homotopy groups of spheres over an algebraically closed field of characteristic zero are determined by -completed spheres and motivic cohomology, enabling the proof that complex realization induces isomorphisms in specific bidegrees and resolving the conditions under which the universal stably-free module of type admits a free summand.
This paper establishes that the filter quotient construction preserves model structures under suitable conditions, demonstrating that while the resulting model inherits properties like being simplicial or proper, it may not remain cofibrantly generated, and further proves its compatibility with filter quotient -categories.
This paper explores invariants of almost embeddings of graphs in the plane by establishing relations among them, connecting these to the homology of the deleted product, constructing examples, and presenting these topological concepts in an accessible manner while highlighting open conjectures.
This paper introduces weighted Grassmann orbifolds via Plücker weight vectors, classifies them up to homeomorphism, establishes conditions for torsion-free integral cohomology, and explicitly computes their integral cohomology rings and equivariant structure constants.
This paper establishes that the decomposition theorem components for contraction maps of torus actions of complexity one are intersection cohomology complexes of even codimensional subvarieties, leading to the vanishing of odd-dimensional intersection cohomology for rational complete varieties of this type and providing explicit formulas for the Betti numbers of affine trinomial hypersurfaces based on their defining equations.
This paper introduces an extended 3-manifold topological model for high-contrast optical flow that resolves the limitations of previous torus-based approaches by identifying that the most significant motion patches are concentrated near binary step-edge circles rather than the torus, thereby offering new insights into the topological and geometric structures underlying visual data inference.
This paper establishes the theorem of the heart for Weibel's homotopy -theory (), proving that the realization functor induces an equivalence for small stable -categories with bounded -structures, a result derived from a strengthened version of Barwick's theorem that provides precise isomorphism ranges for classical -theory and demonstrates the sharpness of these bounds.
This paper investigates the growth rates, asymptotic behaviors, and Gauss congruences of Reidemeister and Nielsen coincidence numbers for endomorphisms of torsion-free nilpotent groups and maps on compact nilmanifolds, while also establishing the rationality of the Nielsen coincidence zeta function.