Simple Connectedness Research Articles

Isabel Fredericson Handlon, PhD (1922–2022)

Isabel Fredericson lived a long and fruitful life, tapping into the hearts of many people. I have the distinction of having met her in kindergarten where we started our schooling by making butter together. We knew each other over the many subsequent years but only on a casual basis, as schoolmates. Then I became more closely connected with her when she entered into the Gestalt Community in Cleveland. There, I came to know her at a very different level, one which honored the sharpness of her mind, the kindness of her spirit, and the promise of new views about life. She was a quiet force, but everybody around her knew that when she said something it was unadorned insight and created a simple connectedness. I remember one time when I was in a group with her and was lamenting the fact that I had little guidance in my growing up years. She said to me, “That may be one of your great assets.” That simple observation was a rich testament to the hidden advantages we grow up with, and I accepted it as a new perspective on what matters.I spent many years of friendship with her and Joseph Handlon, her husband. They were a marvelous example of mutual admiration and simple togetherness. I would love to see her smile once more and to honor the fact that she made me feel the familiarity of coming home again. She did this for many people because the simple feeling of being at home seemed to come very naturally to her. I salute Isabel in our long-time friendship and with the love that colored the spirit of our times together.

Interactions between Homotopy and Topological Groups in Covering (C, R) Space Embeddings

The interactions between topological covering spaces, homotopy and group structures in a fibered space exhibit an array of interesting properties. This paper proposes the formulation of finite covering space components of compact Lindelof variety in topological (C, R) spaces. The covering spaces form a Noetherian structure under topological injective embeddings. The locally path-connected components of covering spaces establish a set of finite topological groups, maintaining group homomorphism. The homeomorphic topological embedding of covering spaces and base space into a fibered non-compact topological (C, R) space generates two classes of fibers based on the location of identity elements of homomorphic groups. A compact general fiber gives rise to the discrete variety of fundamental groups in the embedded covering subspace. The path-homotopy equivalence is admitted by multiple identity fibers if, and only if, the group homomorphism is preserved in homeomorphic topological embeddings. A single identity fiber maintains the path-homotopy equivalence in the discrete fundamental group. If the fiber is an identity-rigid variety, then the fiber-restricted finite and symmetric translations within the embedded covering space successfully admits path-homotopy equivalence involving kernel. The topological projections on a component and formation of 2-simplex in fibered compact covering space embeddings generate a prime order cyclic group. Interestingly, the finite translations of the 2-simplexes in a dense covering subspace assist in determining the simple connectedness of the covering space components, and preserves cyclic group structure.

The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

We study the simple connectedness of the class of finite-dimensional algebras over an algebraically closed field for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. We show that a tame algebra in this class is simply connected if and only if its first Hochschild cohomology space vanishes.

The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander\u2013Reiten components

We study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

On the arithmetic of weighted complete intersections of low degree

A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be rationally connected themselves. We prove that smooth, complex, weighted complete intersections of low enough degree are rationally simply connected. This result has strong arithmetic implications for weighted complete intersections defined over the function field of a smooth, complex curve. Namely, it implies that these varieties satisfy weak approximation at all places, that R-equivalence of rational points is trivial, and that the Chow group of zero cycles of degree zero is zero.

Rogue breather modes: Topological sectors, and the ‘belt-trick’, in a one-dimensional ferromagnetic spin chain

We present explicit solutions for breather soliton modes of excitation in the one-dimensional Heisenberg ferromagnetic spin chain. We identify a characteristic geometrical feature of these breather modes wherein a helicoidal configuration of spins is continuously transformed to one which differs from the initial helicoid by a total twist of `2'. This is a curious manoeuvre popularly known as the `belt trick', an illustration of the simple connectedness of the $\rm{SU}(2)$ group manifold, and its rotation period $4\pi$. We show that this effectively splits the configuration space of the ferromagnetic chain in one-dimension into two topological sectors, distinguished by their total twist -- either `0', or `1'. Further, the energy lower bound of the two sectors is separated by a finite gap varying inversely with the size of the lattice.

Pull-back of Quasi-Log Structures

We prove that the pull-back of a quasi-log scheme by a smooth quasi-projective morphism has a natural quasi-log structure. We treat an application to log Fano pairs. This paper also contains a proof by Kento Fujita of the simple connectedness of log Fano pairs with only log canonical singularities.

HOMOGENEOUS SPACE FIBRATIONS OVER SURFACES

By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface over an algebraically closed field, a variety whose geometric generic fiber is a projective homogeneous space admits a rational point if and only if the elementary obstruction vanishes.

On the universal cover and the fundamental group of an RCD*(K,N)-space

Abstract The main goal of the paper is to prove the existence of the universal cover for 𝖱𝖢𝖣 * ⁢ ( K , N ) {\mathsf{RCD}^{*}(K,N)} -spaces. This generalizes earlier work of [43, 44] on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental group of 𝖱𝖢𝖣 * ⁢ ( K , N ) {\mathsf{RCD}^{*}(K,N)} -spaces. These are the first topological results for 𝖱𝖢𝖣 * ⁢ ( K , N ) {\mathsf{RCD}^{*}(K,N)} -spaces without extra structural-topological assumptions (such as semi-local simple connectedness).

On Simple Connectedness of Minimal Representation-Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

Weakly Modular Graphs and Nonpositive Curvature

This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various `nonpositive curvature' and `local-to-global' properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises in several seemingly-unrelated fields of mathematics: Metric graph theory, Geometric group theory, Incidence geometries and buildings, Theoretical computer science and combinatorial optimization. We give a local-to-global characterization of weakly modular graphs and their subclasses in terms of simple connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With $l_1$-embeddable weakly modular and sweakly modular graphs we associate high-dimensional cell complexes, having several strong topological and geometrical properties (contractibility and the CAT(0) property). Their cells have a specific structure: they are basis polyhedra of even $\triangle$-matroids in the first case and orthoscheme complexes of gated dual polar subgraphs in the second case. We resolve some open problems concerning subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture about CAT(0) metric on the orthoscheme complexes of modular lattices; we answer Chastand's question about prime graphs for pre-median graphs.

Simple connectedness of one space of complex-valued functions

The spaces $\mathbb{C}{\Omega}_n$ have been defined. It has been established that the spaces $\mathbb{C}{\Omega}_n$, $n \geqslant 2$ are simply connected.

Topological Properties of the Catastrophe Map of a General Equilibrium Production Model with Uncertain States of Nature

This paper shows existence and efficiency of equilibria of a production model with uncertainty, where production is modeled in the demand function of the consumer. Existence and efficiency of equilibria are a direct consequence of the catastrophe map being smooth and proper. Topological properties of the equilibrium set are studied. It is shown that the equilibrium set has the structure of a smooth submanifold of the Euclidean space which is diffeomorphic to the sphere implying connectedness, simple connectedness, and contractibility. The set of economies with discontinuous price systems is shown to be of Lebesgue measure zero.

Relatively very free curves and rational simple connectedness

Abstract Rational connectedness is an algebro-geometric analogue of path connectedness depending crucially on the existence of special rational curves called very free curves. For a family of varieties parametrized by a base space, the relevant curves are relatively very free curves. Rational simple connectedness is an algebro-geometric analogue of simple connectedness that depends analogously on the existence of special families of rational curves called very twisting families. Until now, this has been a mysterious notion, frustrating progress. We clarify this by showing that relatively very free curves give very twisting families. We also introduce a new construction of relatively very free curves by studying a natural integrable foliation on the total space of the family. As a consequence, we prove that a general complete intersection in ℙ n is rationally simply connected if and only if it is 2-Fano, i.e., if the first and second graded pieces of the Chern character are positive. This is equivalent to a numerical condition on the type (d 1,...,dc ) of the complete intersection: that d 1 2 + ⋯ + d c 2 ≤ n. This subsumes all previous results on rational simple connectedness of complete intersections. As a corollary, we prove the most general result concerning the Hassett–Tschinkel weak approximation conjecture for 2-Fano complete intersections. Further, combined with a result of de Jong, He and Starr, we prove that every smooth, 2-Fano complete intersection defined over the function field of a surface has a rational point.

The Catastrophe Map of a Two Period Production Model with Uncertainty

This paper shows existence and efficiency of equilibria of a two period production model with uncertainty as a consequence of the catastrophe map being smooth and proper. Its inverse mapping defines a finite covering implying finiteness of equilibria. Beyond the extraction of local equilibrium information of the model, the catastrophe map renders itself well for a global study of the equilibrium set. It is shown that the equilibrium set has the structure of a smooth submanifold of the Euclidean space which is diffeomorphic to the sphere implying connectedness, simple connectedness, and contractibility.

A study of Jacobi–Perron boundary words for the generation of discrete planes

The construction of a Sturmian word, and thus of a discrete line, from a continued fraction development generalizes to higher dimensions. Given any vector v∈R3, a list of 6-connected points approximating the line defined by v may be obtained via a generalized continued fraction algorithm. By duality, a discrete plane with normal vector v can also be generated using a related technique. We focus on such discrete planes, more precisely on the finite patterns generated at each step of the process. We show that the choice of the Jacobi–Perron algorithm as a higher dimension generalization of Euclid’s algorithm together with the specific substitutions deduced from it allows us to guaranty the simple connectedness of those patterns.

An introduction to complex analysis

Part 1 The complex plane: complex numbers open and closed sets in the complex plane limits and continuity. Part 2 Holomorphic function and power series: complex power series elementary functions. Part 3 Prelude to Cauchy's theorem: paths integration along paths connectedness and simple connectedness properties of paths and contours. Part 4 Cauchy's theorem: Cauchy's theorem, level I and II logarithms, argument and index Cauchy's theorem revisited. Part 5 Consequences of Cauchy's theorem: Cauchy's formulae power series representation zeros of holomorphic functions the maximum-modulus theorem. Part 6 Singularities and multifunctions: Laurent's theorem singularities meromorphic functions multifunctions. Part 7 Cauchy's residue theorem: counting zeroes and poles claculation of residues estimation of integrals. Part 8 Applications of contour integration: improper and principal-values integrals integrals involving functions with a finite number of poles and infinitely many poles deductions from known integrals integrals involving multifunctions evaluation of definite integrals. Part 9 Fourier and Laplace tranforms: the Laplace tranform - basic properties and evaluation the inversion of Laplace tranforms the Fourier tranform applications to differential equations proofs of the Inversion and Convolution theorems. Part 10 Conformal mapping and harmonic functions: circles and lines revisited conformal mapping mobius tranformations other mappings - powers, exponentials, and the Joukowski transformation examples on building conformal mappings holomorphic mappings - some theory harmonic functions.

Ratio Numerical Ranges of Operators

Geometric properties of ratio numerical ranges of two Hilbert space operators are studied. Characterizations in terms of algebraic properties of operators are given for connectedness of ratio numerical ranges, and for the situations when the ratio numerical ranges are contained in a circle or in a line. Under a hypothesis of sectoriality (in a weak sense) simple connectedness of ratio numerical ranges is proved, and their boundedness property characterized.

Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

Under suitable hypotheses, we prove that a form of a projective homogeneous variety G/P defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre’s Conjecture II in Galois cohomology for function fields over an algebraically closed field.

Hochschild cohomology of a generalisation of canonical algebras

We compute the Hochschild cohomology groups of a family of finite dimensional algebras, the toupie algebras, characterized by having the same associated ordinary quiver as canonical algebras and any admissible ideal. Moreover, we determine the simple connectedness of them and we give necessary and sufficient conditions for rigidity.