Topological Theorem Research Articles

Two remarks on near-horizon geometries

We show that any extreme black hole with an orthogonally transitive Abelian isometry group has a near-horizon geometry with enhanced symmetry. We also point out a simple proof of the horizon topology theorem of Galloway and Schoen for degenerate horizons.

Second order Volterra-Fredholm functional integrodifferential equations

This paper deals with the study of global existence of solutions to initial value problem for second order nonlinear mixed Volterra-Fredholm functional integrodifferential equations in Banach spaces. The technique used in our analysis is based on an application of the topological transversality theorem known as Leray-Schauder alternative and rely on a priori bounds of solution .

Global existence of bifurcated periodic solutions in a commensalism model with delays

This paper is concerned with a commensalism model with a discrete delay and a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the interior equilibrium is investigated. It is found that the interior equilibrium is asymptotically stable when the discrete delay is less than a certain critical value and unstable when the discrete delay is greater than this critical value. By regarding the delay as the bifurcation parameter, the existence of Hopf bifurcation is also considered. Furthermore, the properties of Hopf bifurcation are determined. In particular, the global existence of bifurcated periodic solutions is established by applying the topological global Hopf bifurcation theorem, which shows that the local Hopf bifurcations imply the global ones after the second critical value of parameter. Finally, numerical simulations supporting the theoretical analysis are also included.

Generalized S-Lemma and strong duality in nonconvex quadratic programming

On the basis of a new topological minimax theorem, a simple and unified approach is developed to Lagrange duality in nonconvex quadratic programming. Diverse generalizations as well as equivalent forms of the S-Lemma, providing a thorough study of duality for single constrained quadratic optimization, are derived along with new strong duality conditions for multiple constrained quadratic optimization. The results allow many quadratic programs to be solved by solving one or just a few SDP’s (semidefinite programs) of about the same size, rather than solving a sequence, often infinite, of SDP’s or linear programs of a very large size as in most existing methods.

Topological representations of matroid maps

The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engstr\"om to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that the process is a functor from the category of matroids with weak maps to the homotopy category of topological spaces. We also give a new and conceptual proof of a result regarding the Whitney numbers of the first kind of a matroid.

The topological Baumgartner-Hajnal theorem

Two new topological partition relations are proved. These are \begin{equation*} \omega _{1} \to (top \alpha +1)^{2}_{k}\end{equation*} and \begin{equation*} \mathbb {R} \to (top \alpha +1)^{2}_{k}\end{equation*} for all $\alpha < \omega _{1}$ and all $k< \omega$. Here the prefix “top” means that the homogeneous set $\alpha +1$ is closed in the order topology. In particular, the latter relation says that if the pairs of real numbers are partitioned into a finite number of classes, there is a homogeneous (all pairs in the same class), well-ordered subset of arbitrarily large countable order type which is closed in the usual topology of the reals. These relations confirm conjectures of Richard Laver and William Weiss, respectively. They are a strengthening of the classical Baumgartner-Hajnal theorem.

Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction

The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars–Schneider IIIb system from a quasi-Hamiltonian reduction of the internally fused double SU(n)×SU(n). In particular, the reduced spectral functions depending respectively on the first and second SU(n) factor of the double engender two toric moment maps on the IIIb phase space CP(n−1) that play the roles of action-variables and particle-positions. A suitable central extension of the SL(2,Z) mapping class group of the torus with one boundary component is shown to act on the quasi-Hamiltonian double by automorphisms and, upon reduction, the standard generator S of the mapping class group is proved to descend to the Ruijsenaars self-duality symplectomorphism that exchanges the toric moment maps. We give also two new presentations of this duality map: one as the composition of two Delzant symplectomorphisms and the other as the composition of three Dehn twist symplectomorphisms realized by Goldman twist flows. Through the well-known relation between quasi-Hamiltonian manifolds and moduli spaces, our results rigorously establish the validity of the interpretation [going back to Gorsky and Nekrasov] of the IIIb system in terms of flat SU(n) connections on the one-holed torus.

Global existence for abstract nonlinear Volterra–Fredholm functional integrodifferential equation

Abstract In the present paper, we investigate the global existence of solutions to initial value problem for nonlinear mixed Volterra–Fredholm functional integrodifferential equations in Banach spaces. The technique used in our analysis is based on an application of the topological transversality theorem known as Leray–Schauder alternative and rely on a priori bounds of solution.

Duality theorems in the Hermitian setting

In this paper we prove the Poincaré Lemma for circulant matrices with C∞ entries and the Mittag–Leffler theorem for Hermitian monogenic matrix functions. We then prove algebraic and topological duality theorems for Hermitian monogenic matrix functions. Finally, we use one of these duality theorems to characterize the C2n-module H(K)′, where H denotes the sheaf of Hermitian monogenic functions. These results extend and clarify those obtained in R. Abreu-Blaya et al. (2012) [1].

The deformation energy and fission barriers of heavy nuclei

The macroscopic part of the fission barriers of medium and heavy nuclei is calculated looking for the minimum of liquid drop energy at fixed volume and elongation without using any shape parameterization. The fission barrier heights, obtained within the Lublin-Strasbourg Drop model, are approximated by a simple analytical expression as functions of Z and (N – Z )/A only. In addition, using the topological theorem by Myers and Świa¸tecki which allows to represent the barrier height as the sum of the ground state microscopic correction (which is calculated and tabulated for all nuclei) and the liquid–drop fission barrier, we propose a simple but quite accurate approximation of the fission barrier heights.

Formalization of a normalization theorem in simplicial topology

In this paper we present a complete formalization of the Normalization Theorem, a result in Algebraic Simplicial Topology stating that there exists a homotopy equivalence between the chain complex of a simplicial set, and a smaller chain complex for the same simplicial set, called the normalized chain complex. Even if the Normalization Theorem is usually stated as a higher-order result (with a Category Theory flavor) we manage to give a first-order proof of it. To this aim it is instrumental the introduction of an algebraic data structure called simplicial polynomial. As a demonstration of the validity of our techniques we developed a formal proof in the ACL2 theorem prover.

Reifenberg parameterizations for sets with holes

We extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization of $E$ by a $d$-dimensional plane or smooth manifold. Such a condition is expressed in terms of square summability for the P. Jones numbers $\beta_1(x,r)$. In particular, it applies in the locally Ahlfors-regular case to provide very big pieces of bi-Lipschitz images of $\R^d$.

Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

We investigate the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and prove that if Mn is simply connected and the k-th Ricci curvature of Mn is bounded below by a quantity involving the mean curvature of Mn and the curvature of the ambient manifold, then Mn is diffeomorphic to the standard sphere \({\mathbb{S}^n}\). For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold Mn is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.

A topological central point theorem

In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.

An extension of Greenberg’s theorem to general valuation rings

We extend Greenberg's strong approximation theorem to schemes of finite presentation over valuation rings with arbitrary value group, using the ultraproduct method of Becker, Denef, Lipshitz and van den Dries. As an application, we prove a closed image theorem (in the strong topology on rational points) for proper morphisms of varieties over valued fields.

A topological index theorem for manifolds with corners

AbstractWe define an analytic index and prove a topological index theorem for a non-compact manifoldM0with poly-cylindrical ends. Our topological index theorem depends only on the principal symbol, and establishes the equality of the topological and analytical index in the groupK0(C*(M)), whereC*(M) is a canonicalC*-algebra associated to the canonical compactificationMofM0. Our topological index is thus, in general, not an integer, unlike the usual Fredholm index appearing in the Atiyah–Singer theorem, which is an integer. This will lead, as an application in a subsequent paper, to the determination of theK-theory groupsK0(C*(M)) of the groupoidC*-algebra of the manifolds with cornersM. We also prove that an elliptic operatorPonM0has an invertible perturbationP+Rby a lower-order operator if and only if its analytic index vanishes.

Characterizing stable complete Erdős space

We focus on the space E ω , the countable infinite power of complete Erd˝ os space Ec. Both spaces are universal spaces for the class of almost zerodimensional spaces. We prove that Eω has the property that it is stable under multiplication with any complete almost zero-dimensional space. We obtain this result as a corollary to topological characterization theorems that we develop for E ω . We also show that σ-compacta are negligible in E ω and that the space is countable dense homogeneous.

A Trotter–Kato type theorem in the weak topology and an application to a singular perturbed problem

In this paper we prove a result of the Trotter–Kato type in the weak topology. Let { A ε } ε > 0 be a family of quasi m-accretive linear operators on a Hilbert space X and let us denote by J λ ε the resolvent of A ε . Under certain conditions, the result states that if for any x ∈ X and k = 1 , 2 , … , the sequence ( J λ ε ) k x converges weakly to ( J λ ) k x as ε → 0 , where J λ is the resolvent of a linear quasi m-accretive operator A on X, then the sequence of the semigroups generated by − A ε tends weakly to the semigroup generated by − A, uniformly with respect to t on compact intervals. The result is different from other results of the same type (see e.g., Yosida (1980) [9, p. 269]) and gives an answer to an open problem put in Eisner and Serény (2010) [3]. It is finally applied to compare the asymptotic behavior of a singular perturbation problem associated to a first order hyperbolic problem with diffusion.

Secondary Chern-Euler forms and the law of vector fields

The Law of Vector Fields is a term coined by Gottlieb for a relative Poincar\'e-Hopf theorem. It was first proved by Morse and expresses the Euler characteristic of a manifold with boundary in terms of the indices of a generic vector field and the inner part of its tangential projection on the boundary. We give two differential-geometric proofs of this topological theorem, in which secondary Chern-Euler forms naturally play an essential role. In the first proof, the main point is to construct a chain away from some singularities. The second proof employs a detailed study of the secondary Chern-Euler form on the boundary, which may be of independent interest. More precisely, we show by explicitly constructing a primitive that, away from the outward and inward unit normal vectors, the secondary Chern-Euler form is exact up to a pullback form. It should be emphasized that we obtain this result in the general case without assuming the metric is locally product near the boundary. In either case, Stokes' theorem is used to complete the proof.

A tight colored Tverberg theorem for maps to manifolds

We prove that any continuous map of an N-dimensional simplex Δ N with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of Δ N to the same point in M: For this we have to assume that N ⩾ ( r − 1 ) ( d + 1 ) , no r vertices of Δ N get the same color, and our proof needs that r is a prime. A face of Δ N is a rainbow face if all vertices have different colors. This result is an extension of our recent “new colored Tverberg theorem”, the special case of M = R d . It is also a generalization of Volovikovʼs 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikovʼs proof, as well as ours, works when r is a prime power.