Wild Embeddings Research Articles

New examples in dimensions N ≥ 4 related to a question of J. W. Cannon and S. G. Wayment

Solving R. J. Daverman’s problem, V. S. Krushkal described sticky Cantor sets in [Formula: see text] for [Formula: see text]. Such sets cannot be isotoped off themselves by small ambient isotopies. Using Krushkal sets, we present a new series of wild embeddings related to a question of J. W. Cannon and S. G. Wayment (1970). Namely, for [Formula: see text], we construct examples of compacta [Formula: see text] with the following two properties: some sequence [Formula: see text] converges homeomorphically to [Formula: see text], but no uncountable family of pairwise disjoint sets [Formula: see text] exists such that each [Formula: see text] is ambiently homeomorphic to [Formula: see text].

On Embedding of the Morse–Smale Diffeomorphisms in a Topological Flow

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse–Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse–Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse–Smale diffeomorphisms to be embedded in topological flows. The progress achieved in solving of Palis’s problem in dimension three is associated with the recently obtained complete topological classification of Morse–Smale diffeomorphisms on threedimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

Hyperbolic groups, 4-manifolds and Quantum Gravity

4-manifolds have special topological properties which can be used to get a different view on quantum mechanics. One important property (connected with exotic smoothness) is the natural appearance of 3-manifold wild embeddings (Alexanders horned sphere) which can be interpreted as quantum states. This relation can be confirmed by using the Turaev-Drinfeld quantization procedure. Every part of the wild embedding admits a hyperbolic geometry uncovering a deep connection between quantum mechanics and hyperbolic geometry. Then the corresponding symmetry is used to get a dimensional reduction from 4 to 2 for infinite curvatures. Physical consequences will be discussed. At the end we will obtain a spacetime representation of a quantum state of geometry by a non-singular fractal space (wild embedding) which is stable in the limit of infinite curvatures.

QUANTUM GEOMETRY AND WILD EMBEDDINGS AS QUANTUM STATES

In this paper, we discuss wild embeddings like Alexanders horned ball and relate them to fractal spaces. We build a C*-algebra corresponding to a wild embedding. We argue that a wild embedding is the result of a quantization process applied to a tame embedding. Therefore, quantum states are directly the wild embeddings. Then we give an example of a wild embedding in the four-dimensional spacetime. We discuss the consequences for cosmology.

TOPOLOGICAL QUANTUM D-BRANES AND WILD EMBEDDINGS FROM EXOTIC SMOOTH ℝ4

This is the next step of uncovering the relation between string theory and exotic smooth ℝ4. Exotic smoothness of ℝ4 is correlated with D6-brane charges in IIA string theory. We construct wild embeddings of spheres and relate them to a class of topological quantum D p-branes as well to KK theory. These branes emerge when there are nontrivial NS–NS H-fluxes on S3 where the topological classes are determined by wild embeddings S2→S3. Then wild embeddings of higher dimensional p-complexes into Sn correspond to D p-branes. These wild embeddings as constructed by using gropes are basic objects to understand exotic smoothness as well Casson handles. Next we build C⋆-algebras corresponding to the embeddings. Finally we consider topological quantum D-branes as those which emerge from wild embeddings in question. We construct an action for these quantum D-branes and show that the classical limit agrees with the Born–Infeld action such that flat branes = usual embeddings.

A Controlled Plus Construction for Crumpled Laminations

Given a closed n-manifold M $(n > 4)$ and a finitely generated perfect subgroup P of ${\pi _1}(M)$, we previously developed a controlled version of Quillen’s plus construction, namely a cobordism (W, M, N) with the inclusion $j:N \mapsto W$ a homotopy equivalence and kernel of ${i_\# }:{\pi _1}(M) \mapsto {\pi _1}(W)$ equalling the smallest normal subgroup of ${\pi _1}(M)$ containing P together with a closed map $p:W \mapsto [0,1]$ such that ${p^{ - 1}}(t)$ is a closed n-manifold for every $t \in [0,1]$ and, in particular, $M = {p^{ - 1}}(0)$ and $N = {p^{ - 1}}(1)$. We accomplished this by constructing an acyclic map of manifolds $f:M \mapsto N$ having the right fundamental groups, and W arose as the mapping cylinder of f with a collar attached along N. The main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when ${\pi _1}(M)$ contains no finitely generated perfect subgroups. By-products of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups.

A controlled plus construction for crumpled laminations

Given a closed n-manifold M ( n > 4 ) (n > 4) and a finitely generated perfect subgroup P of π 1 ( M ) {\pi _1}(M) , we previously developed a controlled version of Quillen’s plus construction, namely a cobordism (W, M, N) with the inclusion j : N ↦ W j:N \mapsto W a homotopy equivalence and kernel of i # : π 1 ( M ) ↦ π 1 ( W ) {i_\# }:{\pi _1}(M) \mapsto {\pi _1}(W) equalling the smallest normal subgroup of π 1 ( M ) {\pi _1}(M) containing P together with a closed map p : W ↦ [ 0 , 1 ] p:W \mapsto [0,1] such that p − 1 ( t ) {p^{ - 1}}(t) is a closed n-manifold for every t ∈ [ 0 , 1 ] t \in [0,1] and, in particular, M = p − 1 ( 0 ) M = {p^{ - 1}}(0) and N = p − 1 ( 1 ) N = {p^{ - 1}}(1) . We accomplished this by constructing an acyclic map of manifolds f : M ↦ N f:M \mapsto N having the right fundamental groups, and W arose as the mapping cylinder of f with a collar attached along N. The main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when π 1 ( M ) {\pi _1}(M) contains no finitely generated perfect subgroups. By-products of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups.

Resolving wild embeddings of codimension-one manifolds in manifolds of dimensions greater than 3

For n⩾4, every embedding of an ( n−1)-manifold in an n-manifold has a δ-resolution for each δ>0. Consequently, for n⩾4, every embedding of an ( n-1)-manifold in an n-manifold can be approximated by tame embeddings.

Wild double tangent ball embeddings of spheres in En

It has been known for years that a 2-sphere Σ in E3 must be flatly embedded if it has double tangent balls on opposite sides of Σ at each of its points. However, when the double tangent balls are not required to be on opposite sides of Σ, pathological embeddings exist. This paper details the allowable wild embeddings of spheres having these indiscrete double tangent balls and discusses the higher dimensional analogues.

Crumpled cubes that are not highly complex

It is proved that any crumpled n-cube C , n ⩾ 5 C,n \geqslant 5 , whose wildness is contained in an ( n − 2 ) (n - 2) -manifold S in Bd C must be of Type 2, which is to say that its wildness, though possibly complicated, is not incredibly complicated.

THE EMBEDDING OF COMPACTA IN EUCLIDEAN SPACE

Recently the fundamental importance of the 1-ULC property of the complementary space in describing a given in En has become clear. Wild embeddings in En are characterized by the absence of the 1-ULC property. In this paper tame and wild embeddings in En of arbitrary compacta in codimension at least 3 are defined. For this purpose the notion of the dimension of embedding of compacta in En is introduced. The main theorem asserts that an of a compactum in En, n≥6, is wild if and only if the complementary space is not 1-ULC. Bibliography: 23 items.