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Olga Frolkina

Publications and source records attributed to Olga Frolkina.

10 recordsLinked to original sources
arXiv2026

On projections of a compact set in $\mathbb R^N$

We apply ideas of geometric measure theory and Baire category theory to topological problems, namely, to topological embeddings of compact sets into Euclidean spaces. In 1947, Borsuk constructed a Cantor set in $\mathbb R^N$, $N\geqslant 3$, such that its projection onto any $(N-1)$-plane contains an $(N-1)$-dimensional ball. This can be strengthened: a desired Cantor set can be obtained from an arbitrary Cantor set by an arbitrarily small isotopy of the space $\mathbb R^N$. The question arises: how do the dimensions of the projections of a compact set $X\subset \mathbb R^N$ behave under a typical ambient isotopy or under a typical ambient homeomorphism? (Typical in the sense of the Baire category.) We solve this problem. As a consequence, we get new criteria of tameness and wildness of a Cantor set in terms of its projections. Our main result strengthens V{ä}isälä's theorem (1979) connecting Hausdorff dimension and Shtan'ko embedding dimension. In its turn, V{ä}isälä's theorem extends results of Nöbeling (1931) and Szpilrajn (1937) on relationship between Hausdorff dimension and topological dimension.

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arXiv2022

Wild high-dimensional Cantor fences in $\mathbb{R}^n$, Part I

Let $\mathcal C$ be the Cantor set. For each $n\geqslant 3$ we construct an embedding $A: \mathcal C \times \mathcal C \to \mathbb R^n$ such that $A(\mathcal C \times \{s\})$, for $s\in\mathcal C$, 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 $n\geqslant 3$ and any non-empty perfect compact set $X$ which is embeddable in $\mathbb R^{n-1}$, we describe an embedding $\mathbb A : X \times \mathcal C \to \mathbb R^n$ such that each $\mathbb A (X \times \mathcal \{s\} )$, $s\in \mathcal C$, contains the corresponding $A (\mathcal C \times \{s\} )$, and is ``nice'' on the complement $\mathbb A (X \times \mathcal \{s\} )-A (\mathcal C \times \{s\} )$; in particular, the images $\mathbb A ( X \times \{s\})$, for $s\in\mathcal C$, 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).

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arXiv2022

Cantor sets with high-dimensional projections

In 1994, J.Cobb constructed a tame Cantor set in $\mathbb R^3$ 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 $\mathbb R^n$, $n\geqslant 3$, 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 $X\subset \mathbb R^n$, $n\geqslant 2$, is a zero-dimensional compactum whose projection into some plane $Π\subset \mathbb R^n$ with $\dim Π\in \{1, 2, n-2, n-1\}$ 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.

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arXiv2022

All projections of a typical Cantor set are Cantor sets

In 1994, John Cobb asked: given $N>m>k>0$, does there exist a Cantor set in $\mathbb R^N$ such that each of its projections into $m$-planes is exactly $k$-dimensional? Such sets were described for $(N,m,k)=(2,1,1)$ by L.Antoine (1924) and for $(N,m,m)$ by K.Borsuk (1947). Examples were constructed for the cases $(3,2,1)$ by J.Cobb (1994), for $(N,m,m-1)$ and in a different way for $(N,N-1,N-2)$ by O.Frolkina (2010, 2019), for $(N,N-1,k)$ by S.Barov, J.J.Dijkstra and M.van der Meer (2012). We show that such sets are exceptional in the following sense. Let $\mathcal C(\mathbb R^N)$ be a set of all Cantor subsets of $\mathbb R^N$ endowed with the Hausdorff metric. It is known that $\mathcal C(\mathbb R^N)$ is a Baire space. We prove that there is a dense $G_δ$ subset $\mathcal P \subset \mathcal C(\mathbb R^N)$ such that for each $X\in \mathcal P$ and each non-zero linear subspace $L \subset \mathbb R^N$, the orthogonal projection of $X$ into $L$ is a Cantor set. This gives a partial answer to another question of J.Cobb stated in the same paper (1994).

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arXiv2022

A new simple family of Cantor sets in $\mathbb{R}^3$ all of whose projections are one-dimensional

In 1994, J.Cobb described a Cantor set in $\mathbb{R}^3$ each of whose projections into 2-planes is one-dimensional. A series of works by other authors developing this field followed. We present another very simple series of Cantor sets in $\mathbb{R}^3$ all of whose projections are connected and one-dimensional. These are self-similar Cantor sets which go back to the work of Louis Antoine, and we celebrate their centenary birthday in 2020-2021.

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arXiv2022

Pairwise disjoint Moebius bands in space

V.V.Grushin and V.P.Palamodov proved in 1962 that it is impossible to place in $R^3$ uncountably many pairwise disjoint polyhedra each homeomorphic to the Moebius band. We generalize this result in two directions. First, we give a generalization of this result to tame subsets in $R^N$, $N\geqslant 3$. Second, we show that in case of $R^3$ the theorem holds for arbitrarily topologically embedded (not necessarily tame) Moebius bands.

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arXiv2022

On a question of B.J. Baker and M. Laidacker concerning disjoint compacta in $\mathbb R^N$

We describe wild embeddings of polyhedra into $\mathbb{R}^N$ which show that the answer to the question of B.J. Baker--M. Laidacker (1989) concerning uncountable families of pairwise disjoint compacta can be twofold. The central idea of our construction is the use of specific wild Cantor sets, namely, Antoine--Blankinship--Ivanov necklaces and Krushkal sticky sets. Our basic tools are Antoine's methods and Shtan'ko demension theory.

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arXiv2022

An answer to a question of J.W. Cannon and S.G. Wayment

Solving R.J. Daverman's problem, V. Krushkal described sticky Cantor sets in $\mathbb R^N$ for $N\geqslant 4$; these sets cannot be isotoped off of itself by small ambient isotopies. Using Krushkal sets, we answer a question of J.W. Cannon and S.G. Wayment (1970). Namely, for $N\geqslant 4$ we construct compacta $X\subset \mathbb R^N$ with the following two properties: some sequence $\{ X_i \subset \mathbb R^N \setminus X, \ i\in\mathbb N \}$ converges homeomorphically to $X$, but there is no uncountable family of pairwise disjoint sets $Y_α\subset \mathbb R^N$ each of which is embedded equivalently to $X$.

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arXiv2009

Minimizing the number of Nielsen preimage classes

We find conditions on topological spaces X, Y and nonempty subset B of Y which guarantee that for each continuous map f from X to Y there exists a map g homotopic to f such that Nielsen preimage classes of g^{-1}(B) are all topologically essential.

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