arXiv subjects
Publications and source records attributed to Arkadiy Skopenkov.
We present short proofs of Tverberg-type theorems for cell complexes by S. Hasui, D. Kishimoto, M. Takeda, and M. Tsutaya. One of them states that for any prime power $r$, any complex $X$ topologically homeomorphic to $S^{(d+1)(r-1)-1}$, and any continuous map $f:X\to\mathbb R^d$ there are pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $X$ such that $f(\sigma_1)\cap\ldots f(\sigma_r)\ne\emptyset$.
The procedure to remove double intersections called the Whitney trick is one of the main tools in the topology of manifolds. The analogues of Whitney trick for $r$-tuple intersections were `in the air' since 1960s. However, only recently they were stated, proved and applied to obtain interesting results. Here we prove and apply the $r$-fold Whitney trick when general position $r$-tuple intersection has positive dimension. A continuous map $f\colon M \to B^d$ from a manifold with boundary to the $d$-dimensional ball is called proper, if $f^{-1}(\partial B^d)=\partial M$. Theorem. Let $D=D_1\sqcup\ldots\sqcup D_r$ be disjoint union of $k$-dimensional disks, and $f:D\to B^d$ a proper map such that $f\partial D_1\cap\ldots\cap f\partial D_r=\emptyset$, and the map $$f^r:\partial(D_1\times\ldots\times D_r)\to (B^d)^r-\{(x,x,\ldots,x)\in(B^d)^r\ :\ x\in B^d\}$$ extends continuously to $D_1\times\ldots\times D_r$. If $rd\ge (r+1)k+3$, then there is a proper map $\bar f:D\to B^d$ such that $\bar f=f$ on $\partial D$ and $\bar fD_1\cap\ldots\cap \bar fD_r=\emptyset$.
A map $φ:K\to R^2$ of a graph $K$ is approximable by embeddings, if for each $\varepsilon>0$ there is an $\varepsilon$-close to $φ$ embedding $f:K\to R^2$. Analogous notions were studied in computer science under the names of cluster planarity and weak simplicity. This short survey is intended not only for specialists in the area, but also for mathematicians from other areas. We present criteria for approximability by embeddings (P. Minc, 1997, M. Skopenkov, 2003) and their algorithmic corollaries. We introduce the van Kampen (or Hanani-Tutte) obstruction for approximability by embeddings and discuss its completeness. We discuss analogous problems of moving graphs in the plane apart (cf. S. Spiez and H. Torunczyk, 1991) and finding closest embeddings (H. Edelsbrunner). We present higher dimensional van Kampen obstruction, its completeness result and algorithmic corollary (D. Repovs and A. Skopenkov, 1998).
A map $f\colon K\to \mathbb R^d$ of a simplicial complex is an almost embedding if $f(σ)\cap f(τ)=\emptyset$ whenever $σ,τ$ are disjoint simplices of $K$. Theorem. Fix integers $d,k\ge2$ such that $d=\frac{3k}2+1$. (a) Assume that $P\ne NP$. Then there exists a finite $k$-dimensional complex $K$ that does not admit an almost embedding in $\mathbb R^d$ but for which there exists an equivariant map $\tilde K\to S^{d-1}$. (b) The algorithmic problem of recognition almost embeddability of finite $k$-dimensional complexes in $\mathbb R^d$ is NP hard. The proof is based on the technique from the Matoušek-Tancer-Wagner paper (proving an analogous result for embeddings), and on singular versions of the higher-dimensional Borromean rings lemma and a generalized van Kampen--Flores theorem.
In this expository note we present a proof of the V.A. Vassiliev conjecture on the planarity of graphs with vertices of degree 4 and certain additional structure. Both statement and proof are accessible to high-school students familiar with basic notions of graph theory. The conjecture was first proved by V.O. Manturov (such a proof was one of the main results of his habilitation thesis). In this note the exposition is made clearer and some comments for beginners are added.
In this expository paper we present short simple proofs of Conway-Gordon-Sachs' theorem on intrinsic linking in three-dimensional space, as well as van Kampen-Flores' and Ummel's theorems on intrinsic intersections. The latter are related to nonrealizability of certain hypergraphs in four-dimensional space. The proofs use a reduction to lower dimensions which allows to exhibit relation between these results. We use elementary language which allows to present the main ideas without technicalities. Thus our exposition is accessible to non-specialists in the area, including students who know basic three-dimensional geometry, and who are ready to learn straightforward four-dimensional generalizations.
We present short proofs of the transcendence of the Liouville and the Mahler numbers. The first proof is known for a long time, the second proof apparently appeared only in 2002-2003. The proofs are accessible to high-school students.
We work in the smooth category. Let N be a closed connected n-manifold and assume that m>n+2. Denote by E^m(N) the set of embeddings N -> R^m up to isotopy. The group E^m(S^n) acts on E^m(N) by embedded connected sum of a manifold and a sphere. If E^m(S^n) is non-zero (which often happens for 2m<3n+4) then no results on this action and no complete description of E^m(N) were known. Our main results are examples of the triviality and the effectiveness of this action, and a complete isotopy classification of embeddings into R^7 for certain 4-manifolds N. The proofs are based on the Kreck modification of surgery theory and on construction of a new embedding invariant. Corollary. (a) There is a unique embedding CP^2 -> R^7 up to isoposition. (b) For each embedding f : CP^2 -> R^7 and each non-trivial knot g : S^4 -> R^7 the embedding f#g is isotopic to f.
Let N be a closed, connected, smooth 4-manifold with H_1(N;Z)=0. Our main result is the following classification of the set E^7(N) of smooth embeddings N->R^7 up to smooth isotopy. Haefliger proved that the set E^7(S^4) with the connected sum operation is a group isomorphic to Z_{12}. This group acts on E^7(N) by embedded connected sum. Boechat and Haefliger constructed an invariant BH:E^7(N)->H_2(N;Z) which is injective on the orbit space of this action; they also described im(BH). We determine the orbits of the action: for u in im(BH) the number of elements in BH^{-1}(u) is GCD(u/2,12) if u is divisible by 2, or is GCD(u,3) if u is not divisible by 2. The proof is based on a new approach using modified surgery as developed by Kreck.