Topological Theorem Research Articles

Symplectic topology and ideal-valued measures

We adapt Gromov’s notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it allows us to discuss three “big fiber theorems”—the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology—from a unified viewpoint. Our main technical tool is an enhancement of the symplectic cohomology theory recently developed by Varolgüneş.

The Machado–Bishop theorem in the uniform topology

The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of C(I×J,X⊗Y) as the closure of C(I,X)⊗C(J,Y) in different senses and the extension of continuous vector-valued functions are studied.

Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds

We connect the Poincaré inequality with the Sobolev inequality on Riemannian manifold in a family of integral inequalities $(1.5)$. For these continuum of inequalities, we obtain topological structure theorems of manifolds generalizing previous unification theorems in both intrinsic and extrinsic settings ([33]). Manifolds which admit any of these integral inequalities are nonparabolic, affect topology, geometry, analysis, and admit nonconstant bounded harmonic functions of finite energy. As a consequence, we have proven a Conjecture of Schoen-Yau ([27, p.74]) to be true in dimension two with hypotheses weaker than that used in [1] and [33]$($ which were weaker than the hypotheses set in the conjecture, $($ cf. Remark 1.5$)$. In the same philosophy and spirit as in ([31]), we prove that if $M$ is a complete $n$-manifold, satisfying $\operatorname{(i)}$ the volume growth condition $(1.1)$, $\operatorname{(ii)}$ Liouville Theorem for harmonic functions, and either $\operatorname{(v)}$ a generalized Poincaré- Sobolev inequality $(1.5)$, or $\operatorname{(vi)}$ a general integral inequality $(1.6)$, and Liouville Theorem for harmonic map $u : M \to K$ with $\operatorname{Sec}^K \le 0$, then $(1)$ $M$ has only one end and $(2)$ there is no nontrivial homomorphism from fundamental group $\pi_1(\partial D)$ into $\pi_1 (K)$ as stated in Theorem $1.5$. Some applications in geometry $(\S 3)$, geometric analysis $(\S 4)$, nonlinear partial differential systems $(\S 5)$, integral inequalities on complete noncompact manifolds $(\S 6)$ are made $($cf. e.g., Theorems $3.1$, $4.1$, $5.1$, and $6.1)$. Whereas we made the first study in ([29, 32]) on how the existence of an essential positive supersolution of a second order partial differential systems $Q(u)=0$ on a Riemannian manifold $M$, (by which we mean a $C^2$ function $v \ge 0$ on $M$ that is positive almost everywhere on $M$, and that satisfies $Q(v)=\operatorname{div}(A(x,v,\nabla v)\nabla v)+b(x,v,\nabla v)v\leq 0\quad (5.1)\, $) affects topology, geometry, analysis and variational problems on the manifold $M$. Whereas we generate the work in [35], under $p$-parabolic stable condition without assuming the $p$-th volume growth condition $\lim _{r \to \infty} r^{-p}\operatorname{Vol}(B_r) =0$. The techniques, concepts, and results employed in this paper can be combined with those of essential positive supersolutions of degenerate nonlinear partial differential systems $($cf. for example, Theorems 5.1 - 5.5, 6.1, etc.$)\, $ generalizing previous work in [32, 4.11], which in term recaptures the work of Schoen-Simon-Yau ([25, Theorem 2]). The combined techniques, concepts and method of [32] and [35] can also be used in other new manifolds we found by an extrinsic average variational method ([34]).

Bounding Radon Numbers via Betti Numbers

Abstract We prove general topological Radon-type theorems for sets in $\mathbb R^{d}$ or on a surface. Combined with a recent result of Holmsen and Lee, we also obtain fractional Helly theorem, and consequently the existence of weak $\varepsilon $-nets as well as a $(p,q)$-theorem for those sets. More precisely, given a family ${\mathcal{F}}$ of subsets of ${\mathbb{R}}^{d}$, we will measure the homological complexity of ${\mathcal{F}}$ by the supremum of the first $\lceil d/2\rceil $ reduced Betti numbers of $\bigcap{\mathcal{G}}$ over all nonempty ${\mathcal{G}} \subseteq{\mathcal{F}}$. We show that if ${\mathcal{F}}$ has homological complexity at most $b$, the Radon number of ${\mathcal{F}}$ is bounded in terms of $b$ and $d$. In case that ${\mathcal{F}}$ lives on a surface and the number of connected components of $\bigcap \mathcal G$ is at most $b$ for any $\mathcal G\subseteq \mathcal F$, then the Radon number of ${\mathcal{F}}$ is bounded by a function depending only on $b$ and the surface itself. For surfaces, if we moreover assume the sets in ${\mathcal{F}}$ are open, we show that the fractional Helly number of $\mathcal F$ is linear in $b$. The improvement is based on a recent result of the author and Kalai. Specifically, for $b=1$ we get that the fractional Helly number is at most three, which is optimal. This case further leads to solving a conjecture of Holmsen, Kim, and Lee about an existence of a $(p,q)$-theorem for open subsets of a surface.

FL* Approach to the Coexistence of Fermi Arcs with Metal–Insulator Crossover in Strongly Underdoped Cuprates

We propose that one can explain the coexistence in the same range of doping and temperature of gapless Fermi arcs with the metal–insulator crossover of in-plane resistivity in strongly underdoped cuprates in terms of the FL* fractionalized Fermi liquid nature of these systems, and that such coexistence is not due simply to disorder effects in the resistivity. The particle excitations of this FL* system derived from variants of the t-J model are the gapless holon carrying charge with small Fermi momentum proportional to the doping, the gapful spinon carrying spin 1/2, and an emergent gauge field coupling them and the hole as a spinon–holon bound state, or more precisely resonance, due to gauge binding, with a Fermi surface respecting the topological Luttinger theorem. In our proposal, Fermi arcs are determined by the hole resonance, whereas the metal–insulator crossover is dominated by spinon–spinon (with subleading holon–holon) gauge interactions, and this dichotomy is able to explain their coexistence.

A Finite Topological Type Theorem for Open Manifolds with Non-negative Ricci Curvature and Almost Maximal Local Rewinding Volume

Abstract In this paper, we present finite topological type theorems for open manifolds with non-negative Ricci curvature, under almost maximal local rewinding volume. Unlike previous related research, our theorems remove the constraints of sectional curvature or conjugate radius, which were crucial additional assumptions on metric regularity in prior results. Notably, our settings do not necessarily satisfy a triangle comparison of Toponogov type. In fact, the method we adopt also extends to many previous related studies.

Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions

Abstract We extend the Becker–Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker–Kechris theorems, as well as Sami’s and Hjorth’s sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms and the equivalence of ‘potentially open’ versus ‘orbitwise open’ Borel sets. We also characterize ‘potentially open’ n-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions and prove a result subsuming Lupini’s Becker–Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures. Our proof method is new even in the classical case of Polish groups and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids and more generally in the point-free context, for open localic groupoids.

Bounds on piercing and line-piercing numbers in families of convex sets in the plane

A family of sets has the (p,q) property if among any p members of it some q intersect. It is shown that if a finite family of compact convex sets in R2 has the (p+1,2) property then it is pierced by ⌊p2⌋+1 lines. A colorful version of this result is proved as well. As a corollary, the following is proved: Let F be a finite family of compact convex sets in the plane with no isolated sets, and let F′ be the family of its pairwise intersections. If F has the (p+1,2) property and F′ has the (r+1,2) property, then F is pierced by (⌊r2⌋2+⌊r2⌋)p points when r⩾2, and by p points otherwise. The proofs use the topological KKM theorem.

Esakia duals of regular Heyting algebras

We investigate in this article regular Heyting algebras by means of Esakia duality. In particular, we give a characterisation of Esakia spaces dual to regular Heyting algebras and we show that there are continuum-many varieties of Heyting algebras generated by regular Heyting algebras. We also study several logical applications of these classes of objects and we use them to provide novel topological completeness theorems for inquisitive logic, DNA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ exttt{DNA}$$\\end{document}-logics and dependence logic.

Updating Utility Functions on Preordered Sets

We consider the problem of extending a function fP defined on a subset P of an arbitrary set X to X strictly monotonically with respect to a preorder ≽ defined on X , without imposing continuity constraints. We show that whenever ≽ has a utility representation, fP is extendable if and only if it is gap-safe increasing. This property means that whenever x′≻x, the infimum of fP on the upper contour of x′ exceeds the supremum of fP on the lower contour of x, where x, x′∈X˜ and X˜ is X completed with two absolute ≽-extrema and, moreover, fP is weakly increasing. The completion of X makes the condition sufficient. The proposed method of extension is flexible in the sense that for any bounded utility representation u of ≽, it provides an extension of fP that coincides with u on a region of X that includes the set of P-neutral elements of X . An analysis of related topological theorems shows that the results obtained are not their consequences. The necessary and sufficient condition of extendability and the form of the extension are simplified when P is a Pareto set.

Baire Category Soft Sets and Their Symmetric Local Properties

In this paper, we study soft sets of the first and second Baire categories. The soft sets of the first Baire category are examined to be small soft sets from the point of view of soft topology, while the soft sets of the second Baire category are examined to be large. The family of soft sets of the first Baire category in a soft topological space forms a soft σ-ideal. This contributes to the development of the theory of soft ideal topology. The main properties of these classes of soft sets are discussed. The concepts of soft points where soft sets are of the first or second Baire category are introduced. These types of soft points are subclasses of non-cluster and cluster soft sets. Then, various results on the first and second Baire category soft points are obtained. Among others, the set of all soft points at which a soft set is of the second Baire category is soft regular closed. Moreover, we show that there is symmetry between a soft set that is of the first Baire category and a soft set in which each of its soft points is of the first Baire category. This is equivalent to saying that the union of any collection of soft open sets of the first Baire category is again a soft set of the first Baire category. The last assertion can be regarded as a generalized version of one of the fundamental theorems in topology known as the Banach Category Theorem. Furthermore, it is shown that any soft set can be represented as a disjoint soft union of two soft sets, one of the first Baire category and the other not of the first Baire category at each of its soft points.

Band-resolved Caroli-de Gennes-Matricon states of multiple-flux-quanta vortices in a multiband superconductor.

Superconductors are of type I or II depending on whether they form an Abrikosov vortex lattice. Although bulk lead (Pb) is classified as a prototypical type-I superconductor, we show that its two-band superconductivity allows for single-flux-quantum and multiple-flux-quanta vortices in the intermediate state at millikelvin temperature. Using scanning tunneling microscopy, the winding number of individual vortices is determined from the real space wave function of its Caroli-de Gennes-Matricon bound states. This generalizes the topological index theorem put forward by Volovik for isotropic electronic states to realistic electronic structures. In addition, the bound states due to the two superconducting bands of Pb can be separately detected and the two gaps close independently inside vortices. This yields strong evidence for a low interband coupling.

Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes

We prove the following quantitative Borsuk–Ulam-type result (an equivariant analogue of Gromov’s Topological Overlap Theorem): Let X be a free ℤ/2-complex of dimension d with coboundary expansion at least ηk in dimension 0 ≤ k < d. Then for every equivariant map F: X →ℤ/2 ℝd, the fraction of d-simplices σ of X with 0 ∈ F (σ) is at least 2−d Πk=0d−1ηk.As an application, we show that for every sufficiently thick d-dimensional spherical building Y and every map f: Y → ℝ2d, we have f(σ) ∩ f(τ) ≠ ∅ for a constant fraction μd > 0 of pairs {σ, τ} of d-simplices of Y. In particular, such complexes are non-embeddable into ℝ2d, which proves a conjecture of Tancer and Vorwerk for sufficiently thick spherical buildings.We complement these results by upper bounds on the coboundary expansion of two families of simplicial complexes; this indicates some limitations to the bounds one can obtain by straighforward applications of the quantitative Borsuk–Ulam theorem. Specifically, we prove• an upper bound of (d + 1)/2d on the normalized (d − 1)-th coboundary expansion constant of complete (d + 1)-partite d-dimensional complexes (under a mild divisibility assumption on the sizes of the parts); and• an upper bound of (d + 1)/2d + ε on the normalized (d − 1)-th coboundary expansion of the d-dimensional spherical building associated with {rm{G}}{{rm{L}}_{d + 2}}({mathbb{F}_q}) for any ε > 0 and sufficiently large q. This disproves, in a rather strong sense, a conjecture of Lubotzky, Meshulam and Mozes.

Covariant star product on semi-conformally flat noncommutative Calabi–Yau manifolds and noncommutative topological index theorem

A differential geometric statement of the noncommutative topological index theorem is worked out for covariant star products on noncommutative vector bundles. To start, a noncommutative manifold is considered as a product space [Formula: see text], wherein [Formula: see text] is a closed manifold, and [Formula: see text] is a flat Calabi–Yau [Formula: see text]-fold. Also, a semi-conformally flat metric is considered for [Formula: see text] which leads to a dynamical noncommutative spacetime from the viewpoint of noncommutative gravity. Based on the Kahler form of [Formula: see text] the noncommutative star product is defined covariantly on vector bundles over [Formula: see text]. This covariant star product leads to the celebrated Groenewold–Moyal product for trivial vector bundles and their flat connections, such as [Formula: see text]. Hereby, the noncommutative characteristic classes are defined properly and the noncommutative Chern–Weil theory is established by considering the covariant star product and the superconnection formalism. Finally, the index of the ⋆-noncommutative version of elliptic operators is studied and the noncommutative topological index theorem is stated accordingly.

Tverberg's theorem for cell complexes

AbstractThe topological Tverberg theorem states that any continuous map of a ‐simplex into the Euclidean ‐space maps some points from pairwise disjoint faces of the simplex to the same point whenever is a prime power. We substantially generalize this theorem to continuous maps of certain CW complexes, including simplicial ‐spheres, into the Euclidean ‐space. We also discuss the atomicity of the Tverberg property.

A topological completeness theorem for transfinite provability logic

We prove a topological completeness theorem for the modal logic textsf{GLP} containing operators {langle xi rangle :xi in textsf{Ord}} intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space X, any sentence phi consistent with textsf{GLP} can be satisfied on a polytopological space based on finitely many Icard topologies constructed over X and corresponding to the finitely many modalities that occur in phi .

Three-Dimensional Graph Products with Unbounded Stack-Number

We prove that the stack-number of the strong product of three n-vertex paths is $$\Theta (n^{1/3})$$ . The best previously known upper bound was O(n). No non-trivial lower bound was known. This is the first explicit example of a graph family with bounded maximum degree and unbounded stack-number. The main tool used in our proof of the lower bound is the topological overlap theorem of Gromov. We actually prove a stronger result in terms of so-called triangulations of Cartesian products. We conclude that triangulations of three-dimensional Cartesian products of any sufficiently large connected graphs have large stack-number. The upper bound is a special case of a more general construction based on families of permutations derived from Hadamard matrices. The strong product of three paths is also the first example of a bounded degree graph with bounded queue-number and unbounded stack-number. A natural question that follows from our result is to determine the smallest $$\Delta _0$$ such that there exists a graph family with unbounded stack-number, bounded queue-number and maximum degree $$\Delta _0$$ . We show that $$\Delta _0\in \{6,7\}$$ .

Topological Explorations on the Fundamental Theorem of Arithmetic

In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and investigate non-standard questions. The ideas are arranged as a series of questions for students to explore, with expositions to the instructor on how to guide them in looking for answers. The investigations culminate in a satisfying outcome: that partition topologies are the same as unique factorization topologies, those that satisfy our topological version of the Fundamental Theorem of Arithmetic.

Stochastic integrodifferential models of fractional orders and Leffler nonsingular kernels: well-posedness theoretical results and Legendre Gauss spectral collocation approximations

Stochastic fractional integrodifferential models are widely employed to model several natural phenomena these days. This current work focuses on the well-posedness results and numerical solutions of a specific form of these models considering the Leffler nonsingular kernels operator wherein the stochastic term is driven by the standard Brownian motion. Accordingly, a combination of sufficient conditions, topological theorems, and Banach space theory are utilized to construct the well-posedness proof. For treating the numerical issue, a familiar spectral collocation technique relying upon shifted Legendre series expansion theory is proposed. The basic properties of Brownian motion and a linear spline interpolation method are used to simulate the standard Brownian motion at a fixed time value. In addition, the idea of the Gauss-Legendre numerical integration rule is implemented to approximate the finite integral. We also devote our attention to the concept of convergence of the proposed method and demonstrate its analysis. Ultimately, the obtained theoretical results and the presented method are examined with five numerous applications. The obtained results indicate the high accuracy and efficiency of applying this method in solving stochastic models of the above-mentioned form.

Topological Equivalence Theorem and Double-Copy for Chern-Simons Scattering Amplitudes.

We study the mechanism of topological mass generation for 3-dimensional Chern-Simons gauge theories and propose a brand-new topological equivalence theorem to connect scattering amplitudes of the physical gauge boson states to that of the transverse states under high-energy expansion. We prove a general energy cancelation mechanism for N-point physical gauge boson amplitudes, which predicts large cancelations of E4 - L → E(4 - L) - N at any L-loop level (L ⩾ 0). We extend the double-copy approach to construct massive graviton amplitudes and to study their structures. We newly uncovered a series of strikingly large energy cancelations E12 → E1 of the tree-level 4-graviton scattering amplitude under high-energy expansion and establish a new correspondence between the 2 energy cancelations in the topologically massive Yang-Mills gauge theory and the topologically massive gravity theory. We further study the scattering amplitudes of Chern-Simons gauge bosons and gravitons in the nonrelativistic limit.