Topological Theorem Research Articles

Topological Censorship for Kaluza–Klein Space-Times

The standard topological censorship theorems require asymptotic hypotheses which are too restrictive for several situations of interest. In this paper we prove a version of topological censorship under significantly weaker conditions, compatible, e.g., with solutions with Kaluza–Klein asymptotic behavior. In particular we prove simple connectedness of the quotient of the domain of outer communications by the group of symmetries for models which are asymptotically flat, or asymptotically anti-de Sitter, in a Kaluza–Klein sense. This allows one, e.g., to define the twist potentials needed for the reduction of the field equations in uniqueness theorems. Finally, the methods used to prove the above are used to show that weakly trapped compact surfaces cannot be seen from Scri.

Optimal shapes and fission barriers of nuclei within the liquid drop model

The fission barriers of medium and heavy nuclei are calculated looking for the minimum of liquid drop energy at fixed volume and elongation without using any shape parametrization. The fission barrier heights, obtained within the Lublin-Strasbourg drop model, are approximated by a simple analytical expression as functions of $Z$ and $(N\ensuremath{-}Z)/A$ only. In addition, using the topological theorem by Myers and \ifmmode \acute{S}\else \'{S}\fi{}wi\k{a}tecki that allows us 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.

Hopf bifurcations in a predator–prey system with multiple delays

This paper is concerned with a two species Lotka–Volterra predator–prey system with three discrete delays. By regarding the gestation period of two species as the bifurcation parameter, the stability of positive equilibrium and Hopf bifurcations of nonconstant periodic solutions are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations (FDEs). In addition, the global existence of bifurcated periodic solutions are also established by employing 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, to verify our theoretical predictions, some numerical simulations are also included.

An Elementary Deduction of the Topological Radon Theorem from Borsuk–Ulam

The Topological Radon Theorem states that, for every continuous function from the boundary of a (d+1)-dimensional simplex into ℝ n , there exists a pair of disjoint faces in the domain whose images intersect in ℝ n . The similarity between that result and the classical Borsuk–Ulam Theorem is unmistakable, but a proof that the Topological Radon Theorem follows from Borsuk–Ulam is not immediate. In this note we provide an elementary argument verifying that implication.

On the topology of untrapped surfaces

Recently a simple proof of the generalizations of Hawking's black hole topology theorem and its application to topological black holes for higher dimensional (n ⩾ 4) spacetimes was given by Rácz I (2008 Class. Quantum Grav. 25 162001). By applying the associated new line of argument it is proven here that strictly stable untrapped surfaces possess exactly the same topological properties as strictly stable marginally outer trapped surfaces (MOTSs) are known to. In addition, a quasi-local notion of outwards and inwards pointing spacelike directions—applicable to untrapped and marginally trapped surfaces—is also introduced.

A Fixed Point Theorem in Weak Topology for Successively Recurrent System of Set-Valued Mapping Equations and Its Applications

Let us introduce n (≥ 2) mappings fi(i = 1, …, n ≡ 0) defined on reflexive real Banach spaces Xi-1 and let fi : Xi-1 → Yi be completely continuous on bounded convex closed subsets $X_{i-1}^{(0)} \\subset X_{i-1}$. Moreover, let us introduce n set-valued mappings $F_i : X_{i-1} imes Y_i o {\\cal F}_c(X_i)$ (the family of all non-empty compact subsets of Xi), (i=1, …, n ≡ 0). Here, we have a fixed point theorem in weak topology on the successively recurrent system of set-valued mapping equations: xi ∈ Fi(xi-1, fi(xi-1)), (i=1, …, n ≡ 0). This theorem can be applied immediately to analysis of the availability of system of circular networks of channels undergone by uncertain fluctuations and to evaluation of the tolerability of behaviors of those systems.

Groupoids and an index theorem for conical pseudo-manifolds

We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M . A main new ingredient in our proof is a non-commutative algebra that plays in our setting the role of 𝒞 0 ( T*M ). We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in K -theory. We then give a new proof of the Atiyah-Singer Index Theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds.

Topological and differentiable sphere theorems for complete submanifolds

Topological and differentiable sphere theorems for complete submanifolds

An application of topological multiple recurrence to tiling

We show that given any tiling of Euclidean space, any geometric pattern of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated version of the pattern. The rather simple proof uses Furstenberg's topological multiple recurrence theorem.

Lattice Operators and Topologies

Working within a complete (not necessarily atomic) Boolean algebra, we use a sublattice to define a topology on that algebra. Our operators generalizecomplementon a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski and Samuel, the operators exhibit some surprising behaviors. We consider properties of such lattices and their interrelations. Many of these properties are abstractions and generalizations of topological spaces. The approach is similar to that of Bachman and Cohen. It is in the spirit of Alexandroff, Frolík, and Nöbeling, although the setting is more general. Proceeding in this manner, we can handle diverse topological theorems systematically before specializing to get as corollaries as the topological results of Alexandroff, Alo and Shapiro, Dykes, Frolík, and Ramsay.

DIGITAL COVERING THEORY AND ITS APPLICATIONS

As a survey-type article, the paper reviews various digital topological utilities from digital covering theory. Digital covering theory has strongly contributed to the calculation of the digital k-fundamental group of both a digital space(a set with k-adjacency or digital k-graph) and a digital product. Furthermore, it has been used in classifying digital spaces, establishing almost Van Kampen theory which is the digital version of van Kampen theorem in algebrate topology, developing the generalized universal covering property, and so forth. Finally, we remark on the digital k-surface structure of a Cartesian product of two simple closed -curves in , .

AN INTEGRAL FORMULA AND ITS APPLICATION TO NORMAL EULER NUMBER

We establish a general integral formula over sphere, and then apply it to give a geometrical proof of the celebrated topological theorem of Lashof and Smale, which asserts that the tangential degree of the tangent sphere bundle coincides with the normal Euler number for an immersion Mn → E2n of an oriented closed manifold into Euclidean space of twice dimension.

VANISHING AND TOPOLOGICAL SPHERE THEOREMS FOR SUBMANIFOLDS IN A HYPERBOLIC SPACE

We extend the vanishing and sphere theorems due to Lawson, Simons, Xin, Shiohama and Xu. By using the techniques of calculus of variations in the geometric measure theory, we prove the vanishing theorem for homology groups of submanifolds in the hyperbolic space Hn(c) with negative constant curvature c. Moreover, we obtain a topological sphere theorem for certain complete submanifolds in Hn(c).

A topological sampling theorem for Robust boundary reconstruction and image segmentation

Existing theories on shape digitization impose strong constraints on admissible shapes, and require error-free data. Consequently, these theories are not applicable to most real-world situations. In this paper, we propose a new approach that overcomes many of these limitations. It assumes that segmentation algorithms represent the detected boundary by a set of points whose deviation from the true contours is bounded. Given these error bounds, we reconstruct boundary connectivity by means of Delaunay triangulation and α -shapes. We prove that this procedure is guaranteed to result in topologically correct image segmentations under certain realistic conditions. Experiments on real and synthetic images demonstrate the good performance of the new method and confirm the predictions of our theory.

Exponential Thurston maps and limits of quadratic differentials

We give a topological characterization of postsingularly finite topological exponential maps, i.e., universal covers g : C → C ∖ { 0 } g\colon \mathbb {C}\to \mathbb {C}\setminus \{0\} such that 0 0 has a finite orbit. Such a map either is Thurston equivalent to a unique holomorphic exponential map λ e z \lambda e^z or it has a topological obstruction called a degenerate Levy cycle. This is the first analog of Thurston’s topological characterization theorem of rational maps, as published by Douady and Hubbard, for the case of infinite degree. One main tool is a theorem about the distribution of mass of an integrable quadratic differential with a given number of poles, providing an almost compact space of models for the entire mass of quadratic differentials. This theorem is given for arbitrary Riemann surfaces of finite type in a uniform way.

Decomposition of effect algebras and the Hammer–Sobczyk theorem

We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., lattice-ordered effect algebras). As a consequence, we obtain a Hammer�Sobczyk type decomposition theorem for modular measures on D-lattices.

On the Number of Topological Types Occurring in a Parameterized Family of Arrangements

Let \({\mathcal{S}}({\mathbb{R}})\) be an o-minimal structure over ℝ, \(T\subset {\mathbb{R}}^{k_{1}+k_{2}+\ell}\) a closed definable set, and $$\pi_{1}:{\mathbb{R}}^{k_{1}+k_{2}+\ell}\rightarrow {\mathbb{R}}^{k_{1}+k_{2}},\qquad \pi_{2}:{\mathbb{R}}^{k_{1}+k_{2}+\ell}\rightarrow {\mathbb{R}}^{\ell},\qquad \pi_{3}:{\mathbb{R}}^{k_{1}+k_{2}}\rightarrow {\mathbb{R}}^{k_{2}}$$ the projection maps as depicted below: For any collection \({\mathcal{A}}=\{A_{1},\ldots,A_{n}\}\) of subsets of \({\mathbb{R}}^{k_{1}+k_{2}}\) , and \(\mathbf{z}\in {\mathbb{R}}^{k_{2}}\) , let \(\mathcal {A}_{\mathbf{z}}\) denote the collection of subsets of \({\mathbb{R}}^{k_{1}}\) $$\{A_{1,\mathbf{z}},\ldots,A_{n,\mathbf{z}}\},$$ where \(A_{i,\mathbf{z}}=A_{i}\cap\pi_{3}^{-1}(\mathbf{z}),\;1\leq i\leq n\) . We prove that there exists a constant C=C(T)>0 such that for any family \({\mathcal{A}}=\{A_{1},\ldots,A_{n}\}\) of definable sets, where each A i =π 1(T∩π −12 (y i )), for some y i ∈ℝℓ, the number of distinct stable homotopy types amongst the arrangements \(\mathcal {A}_{\mathbf{z}},\mathbf{z}\in {\mathbb{R}}^{k_{2}}\) is bounded by \(C\cdot n^{(k_{1}+1)k_{2}}\) while the number of distinct homotopy types is bounded by \(C\cdot n^{(k_{1}+3)k_{2}}.\) This generalizes to the o-minimal setting, bounds of the same type proved in Basu and Vorobjov (J. Lond. Math. Soc. (2) 76(3):757–776, 2007) for semi-algebraic and semi-Pfaffian families. One technical tool used in the proof of the above results is a pair of topological comparison theorems reminiscent of Helly’s theorem in convexity theory. These theorems might be of independent interest in the quantitative study of arrangements.

Leray numbers of projections and a topological Helly-type theorem

Let X be a simplicial complex on the vertex set V. The rational Leray number L (X) of X is the minimal d, such that H ˜ i ( Y ; ℚ ) = 0 for all induced subcomplexes Y ⊂ X and i ⩾ d. Suppose that V = U i = 1 m V i is a partition of V such that the induced subcomplexes X[Vi] are all 0-dimensional. Let π denote the projection of X into the (m − 1)-simplex on the vertex set {1, …, m} given by π(v) = i if v ∈ Vi. Let r = max{|π−1(π(x))|:x ∈ |X|}. It is shown that L ( π ( X ) ) ⩽ r L ( X ) + r − 1 One consequence is a topological extension of a Helly-type result of Amenta. Let F be a family of compact sets in ℝ d such that for any F ′ ⊂ F , the intersection ∩ F ′ is either empty or contractible. It is shown that if G is a family of sets such that for any finite G ′ ⊂ G , the intersection ∩ G ′ is a union of at most r disjoint sets in F, then the Helly number of G is at most r(d + 1).

Tverberg's theorem with constraints

The topological Tverberg theorem claims that for any continuous map of the ( q − 1 ) ( d + 1 ) -simplex σ ( d + 1 ) ( q − 1 ) to R d there are q disjoint faces of σ ( d + 1 ) ( q − 1 ) such that their images have a non-empty intersection. This has been proved for affine maps, and if q is a prime power, but not in general. We extend the topological Tverberg theorem in the following way: Pairs of vertices are forced to end up in different faces. This leads to the concept of constraint graphs. In Tverberg's theorem with constraints, we come up with a list of constraints graphs for the topological Tverberg theorem. The proof is based on connectivity results of chessboard-type complexes. Moreover, Tverberg's theorem with constraints implies new lower bounds for the number of Tverberg partitions. As a consequence, we prove Sierksma's conjecture for d = 2 and q = 3 .

Straightening Theorem for bounded Abelian groups

We prove that every continuous function f with f ( 0 ) = 0 between two bounded Abelian groups G and H equipped with the Bohr topology coincides with a homomorphism when restricted to an infinite subset of the domain. This extends the main results of [K. Kunen, Bohr topology and partition theorems for vector spaces, Topology Appl. 90 (1998) 97–107, D. Dikranjan, S. Watson, A solution to van Douwen's problem on the Bohr topologies, J. Pure Appl. Algebra 163 (2001) 147–158]. Moreover, we give several applications and we answer a question of [B. Givens, K. Kunen, Chromatic numbers and Bohr topologies, Topology Appl. 131 (2) (2003) 189–202].