Applications of almost one-to-one maps

A continuous map of topological spaces X,Y is said to be almost 1-to-1 if the set of the points x∈X such that f⁻¹(f(x))={x} is dense in X; it is said to be light if pointwise preimages are 0-dimensional. In a previous paper we showed that sometimes almost one-to-one light maps of compact and σ-compact spaces must be homeomorphisms or embeddings. In this paper we introduce a similar notion of an almost d-to-1 map and extend the above results to them and other related maps. In a forthcoming paper we use these results and show that if f is a minimal self-mapping of a 2-manifold then point preimages under f are tree-like continua and either M is a union of 2-tori, or M is a union of Klein bottles permuted by f.     

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.     

A solution to de Groot’s absolute cone conjecture

A compactum X is an ‘absolute cone’ if, for each of its points x, the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n-dimensional absolute cone is an n-cell. In this paper, we give a complete solution to that conjecture. In particular, we show that the conjecture is true for n≤3 and false for n≥5. For n=4, the absolute cone conjecture is true if and only if the 3-dimensional Poincaré Conjecture is true.     

Embeddings of homology equivalent manifolds with boundary

We prove a theorem on equivariant maps implying the following two corollaries:(1) Let N and M be compact orientable n-manifolds with boundaries such that M⊂N, the inclusion M→N induces an isomorphism in integral cohomology, both M and N have (n−d−1)-dimensional spines and . Then the restriction-induced map Embm(N)→Embm(M) is bijective. Here Embm(X) is the set of embeddings X→Rm up to isotopy (in the PL or smooth category).(2) For a 3-manifold N with boundary whose integral homology groups are trivial and such that N?D³ (or for its special 2-spine N) there exists an equivariant map , although N does not embed into R³.The second corollary completes the answer to the following question: for which pairs (m,n) for each n-polyhedron N the existence of an equivariant map implies embeddability of N into Rm? An answer was known for each pair (m,n) except (3,3) and (3,2).     

Decompositions into codimension two spheres and approximate fibrations

Let M denote an orientable (n+2)-manifold (n?1) and G an upper semicontinuous decomposition of M into compacta having the shape of the n-sphere. In this context it is shown that the decomposition space is a 2-manifold. Moreover, it is established that the decomposition map is an approximate fibration for n>1, while for n=1 the map is an approximate fibration over the complement of a locally finite set.     

Separation by a codimension-1 map with a normal crossing point

Letf:M ^n–1N ⁿ be an immersion with normal crossings of a closed orientable (n–1)-manifold into an orientablen-manifold. We show, under a certain homological condition, that iff has a multiple point of multiplicitym, then the number of connected components ofN–f(M) is greater than or equal tom+1, generalizing results of Biasi and Romero Fuster (Illinois J. Math. 36 (1992), 500–504) and Biasi, Motta and Saeki (Topology Appl. 52 (1993), 81–87). In fact, this result holds more generally for every codimension-1 continuous map with a normal crossing point of multiplicitym. We also give various geometrical applications of this theorem, among which is an application to the topology of generic space curves.     

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.     

Infinite inflations of crumpled cubes

Based upon recent results characterizing Q-manifolds, this paper sets forth an explicit method for retooling certain pathology arising in finite dimensional manifolds as comparable pathology in the Hilbert cube Q. In particular, with reference to an example constructed by the author and J.J. Walsh, it presents an upper semicontinuous decomposition G of Q into points and a null sequence of cellular arcs such that the associated decomposition space is not a Q-manifold, and it also provides a new procedure for embedding finite dimensional compacta as wild subsets of Q.     

Some topological properties of weakly almost periodic mappings

A map φ:X→X induces a linear operator T:C(X)→C(X) by composition: Tf(x)=f°φ(x). T and φ are termed weakly almost periodic if the sequence {Tⁿ} is precompact in the weak operator topology. Using general structure theorems for weakly almost periodic operators, the properties of these point maps are studied from the viewpoint of dynamical systems. The structure of individual minimal sets and of the union M of all minimal sets of φ are investigated. One key result is that, if X is compact, then φ is a strongly almost periodic (i.e., has uniformly equicontinuous iterates) homeomorphisms of M and M is a retract of X. These and other general results are applied to the case where X is a manifold. Several results in which weak implies strong almost periodicity are obtained.     

Splitting manifolds as M × R where M has a k-fold end structure

Let X be a locally finite simplicial complex of dimension n, n? 5, equipped with a k-fold end structure [4] and consider a piecewise linear (n + 1)-dimensional manifold M that is proper homotopy equivalent to X × R by F:M → X × R, where R is the set of real numbers. The question arises as to whether or not the manifold M can be split, i.e., written as M = N × R where N is a n-manifold and where there is a proper homotopy between F and (p₁ ° F₀) × id:N × R → X × R, preserving the natural (k+1)-fold end structure, where F₀ is F|_N and p₁ is the projection X × R → X. Of particular significance is the fact that X is noncompact. When the construction of such splittings is attempted, algebraic obstructions arise, which vanish if and only if the construction can be completed. This paper develops such an obstruction theory by utilizing methods of L.C. Siebenmann and the k-fold end structures of F. Waldhausen.     

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.     

Separable complete ANR's admitting a group structure are Hilbert manifolds

Let X be a complete-metrizable, separable ANR. The following two facts are shown: (a) if X admits a topological group structure, then either this is a Lie group structure or X is an l₂-manifold; (b) If X is a closed convex set in a complete metric linear space, then X is either locally compact or homeomorphic to l₂.     

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).     

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.     

Division and <Emphasis Type="Italic">k</Emphasis>-th root theorems for <Emphasis Type="Italic">Q</Emphasis>-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z _∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f: K → X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X ² and X × [0, 1] are Q-manifolds as well. We construct a countable family χ of spaces with DDP and cd-AP such that no space X ∈ χ is homeomorphic to the Hilbert cube Q whereas the product X × Y of any different spaces X, Y ∈ χ is homeomorphic to Q. We also show that no uncountable family χ with such properties exists. This work was supported by the Slovenian-Ukrainian (Grant No. SLO-UKR 04-06/07)     

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.     

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.     

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).     

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.