arXiv research

Euler's theorem extended to complex structures.

We are concerned with bifurcation analysis and control of nonlinear Eulerian flows with non-resonant n-tuple Hopf singularity. The analysis is involved with CW complex bifurcations of flow-invariant Clifford hypertori, where we refer to these toral manifolds by toral CW complexes. We observe from primary to tertiary fl…

We prove that if $M$ is a CW-complex, then the homotopy type of the skeletal filtration of $M$ does not depend on the cell decomposition of $M$ up to wedge products with $n$-disks $D^n$, when the later are given their natural CW-decomposition with unique cells of order 0, $(n-1)$ and $n$; a result resembling J.H.C. Whi…

CW-EDMD improves prediction accuracy by learning local Koopman models for different state-space regions.

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…

Defines a simplicial operad related to Fulton-MacPherson.

We determine loop space decompositions of simply-connected four-manifolds, $(n-1)$-connected $2n$-dimensional manifolds provided $n\notin\{4,8\}$, and connected sums of products of two spheres. These are obtained as special cases of a more general loop space decomposition of certain torsion-free $CW$-complexes with wel…

In this note, we introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be ob…

Along with excursions into manifolds with corners, resolution towers of Thom and Whitney stratifications, I show that for a generic gradientlike vector field on a manifold with a Morse function, the stable manifolds give a CW decomposition of the manifold. This has been done before.

Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proo…

To each once-punctured-torus bundle, $T_φ$, over the circle with pseudo-Anosov monodromy $φ$, there are associated two tessellations of the complex plane: one, $Δ(φ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the…

Let X be a subcomplex of the standard CW-decomposition of the n-dimensional torus. We exhibit an explicit optimal motion planning algorithm for X. This construction is used to calculate the topological complexity of complements of general position arrangements and Eilenberg-Mac Lane spaces associated to right-angled Ar…

New method to compute homology and intersection form of 4-manifolds.

Paper introduces fat CW complexes including all closed manifolds.

A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells…

We show a Whitney Approximation Theorem for a continuous map from a manifold to a smooth CW complex. This enables us to show that a topological CW complex is homotopy equivalent to a smooth CW complex in a category of topological spaces. It is also shown that, for any open covering of a smooth CW complex, there exists …

Survey on finite group actions on CW-complexes homotopy to spheres.

For a smoothing Y of a 2-dimensional cyclic quotient singularity X, we construct a simple handle decomposition of Y by using a particular birational map from Y to the projective plane. The manifold Y is built up from the product of an annulus with a disk by attaching 2-handles in a manner which can be described by mean…

Study shows $S^1$ algebraic structure in 2-dimensional CW-complex cobordisms.

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 $x, y\in M$ there is a CW-translation $σ$ such that $σ(x)=y$. We prove that if $F$ is a homogeneous Finsl…

Locally flat submanifolds have finite CW complex complements.

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…

The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real …

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…

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…

The paper introduces optimal transport kernels for comparing cell complexes.

Minimal example found for two finite CW-complexes sharing a common covering.

New calculations of topological complexity for symplectic CW-complexes.

Note on relation between K-cowaist and A-cowaist invariants.

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.

The simplest condition characterizing quasi-finite CW complexes $K$ is the implication $Xτ_h K\implies β(X)τK$ for all paracompact spaces $X$. Here are the main results of the paper: Theorem: If $\{K_s\}_{s\in S}$ is a family of pointed quasi-finite complexes, then their wedge $\bigvee\limits_{s\in S}K_s$ is quasi-fini…

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…

We prove that if $M$ is a CW-complex and $*$ is a 0-cell of $M$, then the crossed module $Π_2(M,M^1,*)$ does not depend on the cellular decomposition of $M$ up to free products with $Π_2(D^2,S^1,*)$, where $M^1$ is the 1-skeleton of $M$. From this it follows that if $G$ is a finite crossed module and $M$ is finite, the…

Simplified proofs for splitting homotopy idempotents.

CW-Gen models improve probabilistic time series forecasting by incorporating prior information.

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…

We study a new bordification of the decorated Teichmüller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify necessary and sufficient conditions for paths in this space of filtered screens to yield …

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 $π_1(X)$ possesses a subnormal series $π_1(X) \rhd G_m \rhd ... \rhd G_0$ with a non-trivial elementary amenable group $G_0$. We investigate the $L^2$-invariants of the universal covering of such a CW-complex $X$. We show that the Novikov-Shubin invarian…

We introduce a novel combinatorial method to study $Q^{**}$-transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplifi…

Groups with special properties always have 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.

Associated to any Coxeter system $(W,S)$, there is a labeled simplicial complex $L$ and a contractible CW-complex $Σ_L$ (the Davis complex) on which $W$ acts properly and cocompactly. $Σ_L$ admits a cellulation under which the nerve of each vertex is $L$. It follows that if $L$ is a triangulation of $\mathbb{S}^{n-1}$,…

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.

Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.

Researchers show a complex structure is not a counterexample to a topological problem.

This article introduces planar ribbons, Vergili ribbon complexes and ribbon nerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topological spaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closure of a pair of nesting, non-concentric filled cycles that includes the boundary but does not …

Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.