Let C be the Cantor set. For each n⩾3 we construct an embedding A:C×C→Rn such that A(C×{s}), for s∈C, are pairwise ambiently incomparable everywhere wild Cantor sets (generalized Antoine's ...necklaces). This serves as a base for another new result proved in this paper: for each n⩾3 and any non-empty perfect compact set X which is embeddable in Rn−1, we describe an embedding A:X×C→Rn such that each A(X×{s}), s∈C, contains the corresponding A(C×{s}), and is “nice” on the complement A(X×{s})−A(C×{s}); in particular, the images A(X×{s}), for s∈C, are ambiently incomparable pairwise disjoint copies of X. This generalizes and strengthens theorems of J.R. Stallings (1960), R.B. Sher (1968), and B.L. Brechner–J.C. Mayer (1988).
In 1994, J. Cobb constructed a tame Cantor set in R3 each of whose projections into 2-planes is one-dimensional. We show that an Antoine's necklace can serve as an example of a Cantor set all of ...whose projections are one-dimensional and connected. We prove that each Cantor set in Rn, n⩾3, can be moved by a small ambient isotopy so that the projection of the resulting Cantor set into each (n−1)-plane is (n−2)-dimensional. We show that if X⊂Rn, n⩾2, is a zero-dimensional compactum whose projection into some plane Π⊂Rn with dimΠ∈{1,2,n−2,n−1} is zero-dimensional, then X is tame; this extends some particular cases of the results of D.R. McMillan, Jr. (1964) and D.G. Wright, J.J. Walsh (1982).
We use the technique of defining sequences which comes back to Louis Antoine.
Branched coverings boast a rich history, ranging from the ramification of Riemann surfaces to the realization of 3-manifolds as coverings branched over knots and spanning both geometric topology and ...algebraic geometry. This work delves into branched coverings “à la Fox” of (
G
,
X
)-manifolds, encompassing three main avenues: Firstly, we introduce a comprehensive class of singular (
G
,
X
)-manifolds, elucidating elementary theory paired with illustrative examples to showcase its efficacy and universality. Secondly, building on Montesinos’ work, we revisit and augment the prevailing knowledge, formulating a Galois theory tailored for such branched coverings. This includes a detailed portrayal of the fiber above branching points. Lastly, we identify local attributes that guarantee the existence of developing maps for singular (
G
,
X
)-manifolds within the branched coverings framework. Notably, we pinpoint conditions that ensure the existence of developing maps for these singular manifolds. This research proves especially pertinent for non-metric singular (
G
,
X
)-manifolds like those of Lorentzian or projective nature, as discussed by Barbot, Bonsante, Suhyoung Choi, Danciger, Seppi, Schlenker, and the author, among others. While examples are sprinkled throughout, a standout application presented is a uniformization theorem “à la Mess” for singular locally Minkowski manifolds exhibiting BTZ-like singularities.
We show that for every sequence
(
n
i
)
, where each
n
i
is either an integer greater than 1 or is
∞
, there exists a simply connected open 3-manifold
M
with a countable dense set of ends
{
e
i
}
so ...that, for every
i
, the genus of end
e
i
is equal to
n
i
. In addition, the genus of the ends not in the dense set is shown to be less than or equal to 2. These simply connected 3-manifolds are constructed as the complements of certain Cantor sets in
S
3
. The methods used require careful analysis of the genera of ends and new techniques for dealing with infinite genus.
Distinguishing Bing-Whitehead Cantor sets GARITY, DENNIS; REPOVŠ, DUŠAN; WRIGHT, DAVID ...
Transactions of the American Mathematical Society,
02/2011, Letnik:
363, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were nonstandard (wild), but still had a simply connected ...complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions greater than three. These Cantor sets in S^{3} if and only if their defining sequences differ by some finite number of Whitehead constructions. As a consequence, there are uncountably many nonequivalent such Cantor sets in S^{3}
We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the ...manifold (which necessarily extends to a homeomorphism of the ends) fixes the ends pointwise. These manifolds are complements of rigid generalized Bing–Whitehead (BW) Cantor sets. Previous examples of rigid Cantor sets with simply connected complement in
had infinite genus and it was an open question as to whether finite genus examples existed. The examples here exhibit the minimum possible genus, genus one. These rigid generalized BW Cantor sets are constructed using variable numbers of Bing and Whitehead links. Our previous result with Željko determining when BW Cantor sets are equivalently embedded in
extends to the generalized construction. This characterization is used to prove rigidity and to distinguish the uncountably many examples.
The chief result implies that an n-manifold S embedded in the interior of an (n + 1)-manifold M as a closed, separating subset is locally flatly embedded if the embedding is well behaved in a locally ...peripheral sense and if S has arbitrarily close neighborhoods Q such that the fundamental groups of appropriate components of$Q \backslash S$admit a uniform finite upper bound on the number of generators.
We point out the sharpness of earlier results of McMillan by exhibiting a map of the $n$-sphere $S^n, n \geqslant 5$, onto itself having acyclic but non-cell-like polyhedra as its nondegenerate point ...inverses and for which the image of the set of nondegenerate point inverses is a Cantor set $K$. Of necessity, $K$ is wildly embedded, and it has the unusual additional property that every self-homeomorphism of $K$ extends to a self-homeomorphism of $S^n$.
On the space of Cantor subsets of R3 Gartside, Paul; Kovan-Bakan, Merve
Topology and its applications,
06/2013, Letnik:
160, Številka:
10
Journal Article
Recenzirano
Odprti dostop
The space of Cantor subsets of R3, denoted C(R3), is a Polish space. We prove this space is path connected and locally path connected. The group of autohomeomorphisms of R3, denoted Aut(R3), acts on ...C(R3) naturally. This action gives us natural invariant classes of Cantor sets and we show that these classes are in the lower levels of the Borel hierarchy, in fact they are open, closed, Fσ or Gδ in C(R3). Moreover, we prove that the classification problem of Cantor sets arising from this action is at least as complicated as the classification of countable linear orders.
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