Embeddability of multiple cones

The main result of this paper is that if X is a Peano continuum such that its nth cone Cn(X) embeds into Rn+2 then X embeds into S². This solves a problem proposed by W. Rosicki.     

Detecting codimension one manifold factors with topographical techniques

We prove recognition theorems for codimension one manifold factors of dimension n?4. In particular, we formalize topographical methods and introduce three ribbons properties: the crinkled ribbons property, the twisted crinkled ribbons property, and the fuzzy ribbons property. We show that X×R is a manifold in the cases when X is a resolvable generalized manifold of finite dimension n?3 with either: (1) the crinkled ribbons property; (2) the twisted crinkled ribbons property and the disjoint point disk property; or (3) the fuzzy ribbons property.     

Laminated decompositions involving a given submanifold

Let M denote a connected (n+1)-manifold (without boundary). We study laminated decompositions of M, by which we mean upper semicontinous decompositions G of M into closed, connected n-manifolds. In particular, given M with a lamination G and N, a locally flat, closed, n-dimensional submanifold, we determine conditions under which M admits another lamination G_N with N?G_N. For n ≠ 3 a sufficient condition is that i: N → M be a homotopy equivalence. For n > 3 we give examples to show that i: N → M being a homology equivalence is not sufficient. We also show how to replace the assumption of local flatness of N with a weaker cellularity criterion (n ? 4) known as the inessential loops condition. We then give examples illustrating the abundance of pathology if M is not assumed to have a preexisting lamination.     

Cellularity in polyhedra

A compact subset X of a polyhedron P is cellular in P if there is a pseudoisotropy of P shrinking precisely X to a point. A proper surjection between polyhedra f:P→Q is cellular if each point inverse of f is cellular in P. It is shown that if f:P→Q is a cellular map and either P or Q is a generalized n-manifold, n≠4, then f is approximable by homeomorphisms. Also, if P or Q is an n-manifold with boundary, n≠4, 5, then a cellular map f:P→Q is approximable by homeomorphisms. A cellularity criterion for a special class of cell-like sets in polyhedra is established.     

Locally G-homogeneous Busemann G-spaces

We present short proofs of all known topological properties of general Busemann G-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally G-homogeneous Busemann G-spaces are homeomorphic and strongly topologically homogeneous. This is a key result in the context of the classical Busemann conjecture concerning the characterization of topological manifolds, which asserts that every n-dimensional Busemann G-space is a topological n-manifold. We also prove that every Busemann G-space which is uniformly locally G-homogeneous on an orbal subset must be finite-dimensional.     

On the homotopy groups of separable metric spaces

The aim of this paper is to discuss the homotopy properties of locally well-behaved spaces. First, we state a nerve theorem. It gives sufficient conditions under which there is a weak n-equivalence between the nerve of a good cover and its underlying space. Then we conclude that for any (n−1)-connected, locally (n−1)-connected compact metric space X which is also n-semilocally simply connected, the nth homotopy group of X, πn(X), is finitely presented. This result allows us to provide a new proof for a generalization of Shelah?s theorem (Shelah, 1988 [18]) to higher homotopy groups (Ghane and Hamed, 2009 [8]). Also, we clarify the relationship between two homotopy properties of a topological space X, the property of being n-homotopically Hausdorff and the property of being n-semilocally simply connected. Further, we give a way to recognize a nullhomotopic 2-loop in 2-dimensional spaces. This result will involve the concept of generalized dendrite which introduce here. Finally, we prove that each 2-loop is homotopic to a reduced 2-loop.     

Detecting codimension one manifold factors with 0-stitched disks

The 0-stitched disks property is introduced and shown to detect codimension one manifold factors of dimension n?4. It is shown that if a space X is an ANR and has the 0-stitched disks property, then X has the disjoint homotopies property. It follows that if a space X is a resolvable generalized manifold of dimension n?4 with the 0-stitched disks property, then X is a codimension one manifold factor. Whether or not the 0-stitched disks property is equivalent to the disjoint homotopies property remains an open question.     

Codimension one spheres in Rn with double tangent balls

In contrast to the situation in $\text{R}$³, where a 2-sphere with double tangent balls at each point must be tamely embedded in $\text{R}$³, there exist wild (n?1)-spheres in $\text{R}$ⁿ for n>3 with this same geometric property. However, if the sphere Σ is tame moduio a subset X that lies in a polyhedron P that is tame in Σ, the dimension of P is less than n?2, n>4, and Σ has double tangent balls over X, then Σ must be tame in $\text{R}$ⁿ. Also if the tangent balls extend over P and are pairwise congruent, the dimensional restriction on P can be dropped. Examples are given to support the necessity of the hypotheses of the included theorems.     

Higher-dimensional Bruckner-Garg type theorem

In this paper, we investigate several properties of maps from a compactum X to an n-dimensional (combinatorial) manifold Mn. We introduce the notions of stable point and locally extreme point of map, and we prove a higher-dimensional Bruckner-Garg type theorem for the fiber structure of a generic map in the space C(X,Mn) of maps from a compactum X with dimX?n to an n-dimensional manifold Mn (n?1). As applications, we also study the spaces of Bing maps, Lelek maps, k-dimensional maps and Krasinkiewicz maps in C(X,Mn).     

On higher analogs of topological complexity

Farber introduced a notion of topological complexity TC(X) that is related to robotics. Here we introduce a series of numerical invariants TCn(X), n=2,3,… , such that TC₂(X)=TC(X) and TCn(X)?TCn+1(X). For these higher complexities, we define their symmetric versions that can also be regarded as higher analogs of the symmetric topological complexity.     

The behavior on fundamental group of a free pro-homotopy equivalence

We study the question: given a morphism $\text{?{(X}{}_{\mathrm{n}}\text{, x}{}_{\mathrm{n}}\text{)}→{(Y}{}_{\mathrm{n}}\text{, y}{}_{\mathrm{<mtext>n</mtext>}}\text{)}}$ in the category pro-(Poi nted. Homotopy) where the domain and range are inverse sequences of well-pointed CW complexes, and given that ? induces an isomorphism {X_n}→{Y_n} in pro-(Homotopy), what additional hypotheses force ? to be an isomorphism in pro-(Pointed Homotopy)? Conjecture. If the dimensions of the Y_n's are bounded, then ? is an isomorphism in pro-(Pointed Homotopy). We first prove the special case of this conjecture in which dim Y_n?d<∞ for all n, and $\text{lim}\text{{H}{}_{\mathrm{d}}\text{Y}{}_{\mathrm{n}}\text{}≠0}$, Y_n being the universal cover of Y_n. Then we deal with the general case: we show that there are certain elements of each π₁Y_n with the properties: (i) these elements commute if and only if ? is an isomorphism in pro-(Pointed Homotopy); (ii) if dim Y_n?d<∞ for all n, then powers of these elements commute. While (i) and (ii) are not incompatible, this result puts severe restrictions on the nature of any possible counter-example to the conjecture.Our method also gives pro-homotopy analogues of the well-known fact that if a K(π, 1) is N-dimensional, then π is torsion-free and contains no abelian subgroup of rank>N. The latter theorems apply to inverse sequences {Y_n} of CW complexes where dim Y_n is finite but not necessarily bounded, hence in particular to infinite-dimensional shape-aspherical compacta.     

o-Boundedness of free topological groups

Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F(X) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle fin_?(O,Ω) (the latter means that for every sequence 〈un〉_n∈ω of open covers of T there exists a sequence 〈vn〉_n∈ω such that vn∈[un]^<ω and for every F∈[X]^<ω there exists n∈ω with F⊂?vn). This characterization gives a consistent answer to a problem posed by C. Hernándes, D. Robbie, and M. Tkachenko in 2000.     

Characterizing filters by convergence (with respect to filters) in Banach spaces

Let X be a topological space and let F be a filter on N, recall that a sequence (xn)_n∈N in X is said to be F-convergent to the point x∈X, if for each neighborhood U of x, {n∈N:xn∈U}∈F. By using F-convergence in ?₁ and in Banach spaces, we characterize the P-filters, the P-filters⁺, the weak P-filters, the Q-filters, the Q-filters⁺, the weak Q-filters, the selective filters and the selective⁺ filters.     

Postulation of disjoint unions of lines and a few planes

Let X⊂Pⁿ, n≥5, be a general union of x planes and y lines. Here we prove that X has the expected postulation if x is small with respect to y.     

Tangent cones and C1 regularity of definable sets

Let $X?{\mathbb{R}}^n$ be a connected locally closed set which is definable in an o-minimal structure. We prove that the following three statements are equivalent: (i) X is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of X coincide at every point in X, (iii) for every $x\in X$, the tangent cone of X at the point x is a k-dimensional linear subspace of ${\mathbb{R}}^n$ (k does not depend on x) varies continuously in x, and the density $\theta (X,x)<3/2$.     

An approximation theorem in shape theory

In this paper it is shown that if X is a compactum in the interior of a PL manifold M and if U is a neighborhood of X in M, then there is a compactum X′ in U such that X and X′ have the same relative shape in U and the embedding dimension of X′ equals the fundamental dimension of X. Whenever the dimension of M is not equal to three, the relative shape equivalence from X′ to X can be realized by an infinite isotopy of M.     

Hyperbolicity of arborescent tangles and arborescent links

In this paper, we study the hyperbolicity of arborescent tangles and arborescent links. We will explicitly determine all essential surfaces in arborescent tangle complements with non-negative Euler characteristic, and show that given an arborescent tangle T, the complement X(T) is non-hyperbolic if and only if T is a rational tangle, T=Qm*T^′ for some m?1, or T contains Qn for some n?2. We use these results to prove a theorem of Bonahon and Siebenmann which says that a large arborescent link L is non-hyperbolic if and only if it contains Q₂.     

On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds

Let (X,L) be a polarized manifold of dimension n defined over the field of complex numbers. In this paper, we treat the case where n=3 and 4. First we study the case of n=3 and we give an explicit lower bound for h⁰(K_X+L) if κ(X)≥0. Moreover, we show the following: if κ(K_X+L)≥0, then h⁰(K_X+L)>0 unless κ(X)=−∞ and h¹(O_X)=0. This gives us a partial answer of Effective Non-vanishing Conjecture for polarized 3-folds. Next for n=4 we investigate the dimension of H⁰(K_X+mL) for m≥2. If n=4 and κ(X)≥0, then a lower bound for h⁰(K_X+mL) is obtained. We also consider a conjecture of Beltrametti-Sommese for 4-folds and we can prove that this conjecture is true unless κ(X)=−∞ and h¹(O_X)=0. Furthermore we prove the following: if (X,L) is a polarized 4-fold with κ(X)≥0 and h¹(O_X)>0, then h⁰(K_X+L)>0.     

Blockers in hyperspaces

Given a metric continuum X, let X² and C(X) denote the hyperspaces of all nonempty closed subsets and subcontinua, respectively. For A,B∈X² we say that B does not block A if A∩B=∅ and the union of all subcontinua of X intersecting A and contained in X−B is dense in X. In this paper we study some sets of blockers for several kinds of continua. In particular, we determine their Borel classes and, for a large class of locally connected continua X, we recognize them as cap-sets.