Paper introduces fat CW complexes including all closed manifolds.
problem No specific problem stated, focuses on introducing new CW complexes.
method Introduces a new smooth version of CW complexes called fat CW complexes.
result Fat CW complexes include all closed manifolds and have desirable properties.
Study shows S1 algebraic structure in 2-dimensional CW-complex cobordisms.
problem Characterize cobordisms of 2-dimensional CW-complexes.
method Algebraic characterisation using Hopf algebras and symmetric monoidal categories.
result Category of cobordisms is equivalent to a freely generated Hopf algebra.
The paper introduces optimal transport kernels for comparing cell complexes.
problem Lack of machine learning methods for CW complexes.
method Derives explicit expression for Wasserstein distance, extends Fused Gromov-Wasserstein, introduces novel kernels.
result Introduced novel kernels for comparing probability measures on CW complexes.
Euler's theorem extended to complex structures.
problem Generalizing Euler's theorem to complex structures.
method Analyzing strongly connected, pure n-dimensional regular CW-complexes. result Evenness of cells is equivalent to generalized cycle decomposition and traversability.
This paper proves some results on negative gradient dynamics of Morse functions on Hilbert manifolds. It contains the compactness of flow lines, manifold structures of certain compacti- fied moduli spaces, orientation formulas, and CW structures of the underlying manifolds.
For any compact Lie group G we discuss the relation of the equivariant Reidemeister and analytic torsion of G-manifolds with their G-CW structures.
Study controls bifurcations in Eulerian flows with multiple Hopf singularities.
problem Bifurcation analysis and control of nonlinear Eulerian flows with non-resonant n-tuple Hopf singularities.
method Analysis of CW complex bifurcations of flow-invariant Clifford hypertori, using leaf-bifurcation varieties.
result Tertiary toral CW complex bifurcates from and persists outside a secondary toral CW complex.
The paper shows how to approximate continuous maps to smooth CW complexes.
problem Approximating continuous maps to smooth CW complexes.
method Whitney Approximation Theorem for continuous maps to smooth CW complexes.
result Topological CW complexes are homotopy equivalent to smooth CW complexes.
Researchers show a complex structure is not a counterexample to a topological problem.
problem Wall's D2 problem about finite CW-complexes.
method Introduced and analyzed new presentations of quaternion groups to prove homotopy types.
result The complex structure is not a counterexample to Wall's D2 problem.
Let n be a natural number equal or greater than 2. In this paper we study the topological structure of certain hyperspaces of convex subsets of constant width, equipped with the Hausdorff metric topology. We focus our attention on the hyperspace cw_D(R^n) of all compact convex subsets with constant width d\in D, where …
Survey on finite group actions on CW-complexes homotopy to spheres.
problem Understanding finite group actions on CW-complexes homotopy equivalent to spheres.
method Survey of extensive literature on finite G-CW-complexes homotopy equivalent to spheres. result Finite G-CW-complexes homotopy equivalent to spheres have finite group actions. An isometry of a Finsler space is called Clifford-Wolf translation (CW-translation) if it moves all points the same distance. A Finsler space (M,F) is called Clifford-Wolf homogeneous (CW-homogeneous) if for any x,y∈M there is a CW-translation σ such that σ(x)=y. We prove that if F is a homogeneous Finsl…
Self-affine tiles homeomorphic to a ball proven for a specific digit set.
problem Topology of self-affine tiles with collinear digit sets.
method Proving homeomorphism to a ball using integral self-affine tiles with collinear digit sets.
result A large class of integral self-affine tiles with collinear digit sets is homeomorphic to a closed 3-dimensional ball.
In a previous paper, under the assumption that the Riemannian metric is special, the author proved some results about the moduli spaces and CW structures arising from Morse theory. By virtue of topological equivalence, this paper extends those results by dropping the assumption on the metric. In particular, we give a s…
Locally flat submanifolds have finite CW complex complements.
problem Understanding the structure of manifold complements.
method Direct proof using homotopy equivalence and CW complexes.
result Complements of locally flat submanifolds are finite CW complexes.
We show that the Prüfer surface, which is a separable non-metrizable 2-manifold, has not the homotopy type of a CW-complex. This will follow easily from J. H. C. Whitehead's result: if one has a good approximation of an arbitrary space by a CW-complex, which fails to be a homotopy equivalence, then the given space is n…
It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical point…
This paper formalizes manifolds in positive characteristic varieties.
problem Establishing l-adic formal manifold structures on positive characteristic varieties.
method Develops and proves the existence of l-adic formal manifold structures and abelianized Galois symmetries.
result Proves l-adic homotopic equivalence and l-local lifting for simply-connected varieties.
We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regu…
New calculations of topological complexity for symplectic CW-complexes.
problem Calculating topological complexity for symplectic CW-complexes.
method Using atoroidal cohomology classes and CW-complexes, proving topological complexity for symplectic spaces.
result Every atoroidally symplectic CW-complex of dimension 2n has topological complexity 4n.
Minimal example found for two finite CW-complexes sharing a common covering.
problem Finding the minimum number of 2-cells in two finite CW-complexes that share a common covering.
method Constructing an almost minimal example with two 2-cells in each complex.
result Minimal example with two 2-cells in each complex.
Paper constructs a complex from flow data, capturing manifold's structure.
problem Understanding flows on manifolds with boundaries.
method Origami map from disk to CW-complex, reconstructing manifold's topology.
result Compact CW-complex homotopy equivalent to manifold X. Note on relation between K-cowaist and A-cowaist invariants.
problem Relation between K-cowaist and A-cowaist invariants on manifolds.
method Detailed proof of inequality involving these invariants.
result Established inequality K-cw2(M) ≤ c * A-cw2(M).
New method refines Morse theory for group presentations.
problem Studying transformations of group presentations.
method Refined discrete Morse theory for CW-complexes.
result Some counterexamples to the Andrews--Curtis conjecture are shown to satisfy the conjecture.
We describe a smooth structure, called Frölicher space, on CW complexes and spaces of triangulations. This structure enables differential methods for e.g. minimization of functionnals. As an application, we exhibit how an optimized triangulation can be obtained in order to solve a standard PDE.
The simplest condition characterizing quasi-finite CW complexes K is the implication XτhK⟹β(X)τK for all paracompact spaces X. Here are the main results of the paper: Theorem: If {Ks}s∈S is a family of pointed quasi-finite complexes, then their wedge s∈S⋁Ks is quasi-fini…
We construct for an equivariant cohomology theory for proper equivariant CW-complexes an equivariant Chern character, provided that certain conditions about the coefficients are satisfied. These conditions are fulfilled if the coefficients of the equivariant cohomology theory possess a Mackey structure. Such a structur…
The paper introduces vortex nerve complexes and new Betti numbers in CW spaces.
problem Understanding the structure and properties of CW complexes and their nerves.
method Introducing vortex nerve complexes and defining new Betti numbers for CW complexes.
result New Betti numbers (vortex Bvtex, vortex nerve BvNrv, shape Bsh) are introduced and studied. CW Networks leverage cell complexes to enhance GNNs, achieving state-of-the-art results on molecular datasets.
problem Graph Neural Networks struggle with long-range interactions and lack principled ways to model higher-order structures.
method CW Networks use cell complexes to decouple computational and input graph structures, enabling flexible hierarchical message passing.
result CW Networks achieve state-of-the-art results on molecular datasets.
This paper defines ribbons and ribbon complexes in CW spaces and analyzes their topological properties.
problem Characterizing and analyzing topological structures in CW spaces.
method Introducing planar ribbons, ribbon complexes, and ribbon nerves in Alexandroff-Hopf-Whitehead CW spaces, and studying their topological properties.
result Characterization of ribbons and ribbon nerves by Betti numbers and homotopy types.
Totally nonnegative flag varieties are shown to be regular CW complexes.
problem The structure of totally nonnegative flag varieties.
method Confirmation through CW complex structure and homeomorphism of positroid cells.
result Totally nonnegative flag varieties are regular CW complexes.
Let (M,F) be a connected Finsler space. An isometry of (M,F) is called a Clifford-Wolf translation (or simply CW-translation) if it moves all points the same distance. The compact Finsler space (M,F) is called restrictively Clifford-Wolf homogeneous (restrictively CW-homogeneous) if for any two sufficiently close…
The article defines hyperconnected relator spaces and their properties.
problem Understanding the nearness of path-connected sub-complexes in CW spaces.
method Introduces hyperconnectedness and applies it to CW complexes and continuous functions.
result Existence of continuous functions that are paths in hyperconnected relator spaces.
CW-Gen models improve probabilistic time series forecasting by incorporating prior information.
problem Challenges in probabilistic forecasting of multivariate time series due to non-stationarity, inter-variable dependencies, and distribution shifts.
method CW-Gen framework that incorporates prior information through conditional whitening. JMCE learns conditional mean and covariance, improving sample quality.
result CW-Gen consistently enhances predictive performance, capturing non-stationary dynamics and inter-variable correlations more effectively than prior-free approaches.
Simplified proofs for splitting homotopy idempotents.
problem Understanding the splitting of homotopy idempotents.
method Simplified proofs for both pointed and unpointed cases.
result Homotopy idempotents split in specific categories.
We consider an embedding of a 2-dimensional CW complex into the 3-sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the 2-dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equa…
CW-ICA improves on ANICA for non-linear source separation.
problem Non-linear source separation challenges with many applications.
method CW-ICA extends ANICA by using a simpler, closed-form optimization target.
result CW-ICA achieves comparable results to ANICA without adversarial training.
We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they ge…
Let X be a finite aspherical CW-complex whose fundamental group π1(X) possesses a subnormal series π1(X)⊳Gm⊳...⊳G0 with a non-trivial elementary amenable group G0. We investigate the L2-invariants of the universal covering of such a CW-complex X. We show that the Novikov-Shubin invarian…
CW's time change models for option pricing are flawed.
problem CW's models for time changes in option pricing are not measurable.
method Analysis of the measurability of time changes with respect to the underlying filtration.
result CW's models for time changes fail to satisfy the measurability assumption.
Defines a simplicial operad related to Fulton-MacPherson.
problem Equipping the symplectic cochain complex with BV algebra structure.
method Constructs an operad in Top with CW and simplicial decompositions.
result Expected isomorphism to the 2-dimensional Fulton-MacPherson operad.
CW normalizes and decorrelates neural network layers for better concept understanding.
problem Challenges in understanding neural network hidden layers.
method Introducing concept whitening (CW) to alter neural network layers.
result CW provides clearer understanding of how networks learn concepts over layers.
Groups with special properties always have fixed points.
problem Groups acting on finite CW-complexes without fixed points.
method Exhibited specific groups with strong fixed-point properties.
result Groups with finite generation and torsion-freeness have global fixed points.
We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.
Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.
problem Understanding and characterizing one-dimensional non-Hausdorff manifolds.
method Analyzing properties of connected non-Hausdorff manifolds and their quotient spaces to CW complexes.
result Existence of a quotient map from a connected non-Hausdorff manifold to an open one-dimensional CW complex.
Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.
problem Computing Novikov-Shubin invariants for complex cell structures.
method Construct random walks on cell complexes, relate to Laplacians, and use return probabilities.
result Novikov-Shubin invariants can be recovered from random walk return probabilities.
Two complexes share a common covering but not a finite one.
problem Common covering of complexes with specific properties.
method Constructing CW-complexes K and L with a common covering.
result K is homeomorphic to a complex with a single 2-cell.
The paper proves new results on Poincaré duality pairs and spaces.
problem Establishing Poincaré duality in various contexts and dimensions.
method Analyzes Poincaré spaces and CW pairs, proving relative Poincaré duality and related results.
result Found a finite CW pair (X,Y) where Y fails to satisfy Poincaré duality in any dimension.