arXiv subjects

Sergey A. Melikhov

Publications and source records attributed to Sergey A. Melikhov.

At least 19 recordsLinked to original sources

arXiv2026

Satellites and invariants of links

An invariant $v$ of $m$-component links is called "cableable" if there exists a $k$ such that whenever a link $L'$ is obtained from a link $L=(K_1,\dots,K_m)$ by replacing each knot $K_i$ with its $(p_i,q_i)$-cable for some $p_i$ and $q_i$, we have $v(L')=(p_1\cdots p_m)^kv(L)$. The following problem is implicit in a number of papers by P. M. Akhmetiev and originates from the Arnold-Moffatt program for finding topological lower bounds for the energy of a magnetic field: Does there exist a cableable finite type invariant of links in $S^3$ which is not a function of the pairwise linking numbers? A potential solution of this problem was proposed by Akhmetiev himself, with the desired invariant defined as an analytic expression involving a magnetic field modeled on the given link, but we note that basic properties that he claimed of his invariant cannot be all true. In any case, we offer a different solution, with the desired invariant being a function of the coefficients of the Conway polynomial of the link and its sublinks. Moreover, we show that the cables can be replaced by arbitrary satellites. Much of the proof is a study of low degree coefficients of the Conway potential function $Ω_L(x_1,\dots,x_n)$ expanded as a formal power series in Conway's variables $z_i=x_i-x_i^{-1}$. We also discuss type $n$ invariants which are "cableable up to an invariant of type $n-1$", some cableable invariants which are not of finite type (particularly a certain modification of Milnor's $\barμ$-invariants), and applications to links of solenoids.

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arXiv2025

Lifting generic maps to embeddings. The double point obstruction

Given a generic PL map or a generic smooth fold map $f:N^n\to M^m$, where $m\ge n$ and $2(m+k)\ge 3(n+1)$, we prove that $f$ lifts to a PL or smooth embedding $N\to M\times\mathbb R^k$ if and only if its double point locus $\{(x,y)\in N\times N\mid f(x)=f(y),\,x\ne y\}$ admits an equivariant map to $S^{k-1}$. As a corollary we answer a 1990 question of P. Petersen and obtain some other applications. We also discuss several criteria for lifting of a non-degenerate PL map or a $C^0$-stable smooth map $f:N^n\to M^m$, where $m\ge n$, to an embedding in $M\times\mathbb R$, elaborating on V. Poénaru's observations. In particular, the existence of such a lift is determined by the equivariant homotopy type of the diagram consisting of the three projections from the triple point locus $\{(x,y,z)\in N\times N\times N\mid f(x)=f(y)=f(z),\,x\ne y\ne z\ne x\}$ to the double point locus. The three Appendices, which can be read independently of the rest of the paper, are devoted to stable and generic maps. Appendix B introduces an elementary theory of stable PL maps. Appendix C extends the 2-multi-0-jet transversality theorem over the usual compactification of $M\times M\setminusΔ_M$.

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arXiv2025

A joint logic of problems and propositions

In a 1985 commentary to his collected works, Kolmogorov informed the reader that his 1932 paper 'On the interpretation of intuitionistic logic' "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types - propositions and problems." We construct such a formal system as well as its predicate version, QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The axioms of QHC are obtained as a result of a simultaneous formalization of two well-known alternative explanations of intiuitionistic logic: 1) Kolmogorov's problem interpretation (with familiar refinements by Heyting and Kreisel) and 2) the proof interpretation by Orlov and Heyting, as clarified and extended by Gödel.

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arXiv2025

Topological isotopy and finite type invariants

In 1974, D. Rolfsen asked: If two PL links in $S^3$ are isotopic (=homotopic through embeddings), then are they PL isotopic? We prove that they are PL isotopic to another pair of links which are indistinguishable from each other by finite type invariants. Thus if finite type invariants separate PL links in $S^3$, then Rolfsen's problem has an affirmative solution. In fact, we show that finite type invariants separate PL links in $S^3$ if and only if Rolfsen's problem has an affirmative solution and certain 5 other (rather diverse) conjectures hold simultaneously. We also show that if $v$ is a finite type invariant (or more generally a colored finite type invariant) of PL links, and $v$ is invariant under PL isotopy, then $v$ assumes the same value on all sufficiently close $C^0$-approximations of any given topological link; moreover, the extension of $v$ by continuity to topological links is an invariant of isotopy. Some specific invariants of this kind are discussed.

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arXiv2025

Local knots and the prime factorization of links

The present note contains a new proof of Y. Hashizume's 1958 theorem that every non-split link in $S^3$ admits a unique factorization into prime links. While the new proof does not go far beyond standard techniques, it is considerably shorter than the original proof and avoids most of its case exhaustion. We apply this proof to obtain a string link version (and also an alternative proof) of a 1972 theorem of D. Rolfsen: two PL links in $S^3$ are ambient isotopic if and only if they are PL isotopic and their respective components are ambient isotopic. It is tempting to dismiss this string link version as obvious by deriving it directly either from Rolfsen's or Hashizume's theorem. But this does not seem to be possible, as it turns out that there exists a string link that has no local knots, while its closure has a local knot.

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arXiv2024

Is every knot isotopic to the unknot?

In 1974, D. Rolfsen asked: Is every knot in $S^3$ isotopic (=homotopic through embeddings) to a PL knot or, equivalently, to the unknot? In particular, is the Bing sling isotopic to a PL knot? We show that the Bing sling is not isotopic to any PL knot: (1) by an isotopy which extends to an isotopy of $2$-component links with $lk=1$; (2) through knots that are intersections of nested sequences of solid tori. There are also stronger versions of these results. In (1), the additional component may be allowed to self-intersect, and even to get replaced by a new one as long as it represents the same conjugacy class in $G/[G',G'']$, where $G$ is the fundamental group of the complement to the original component. In (2), the "solid tori" can be replaced by "boundary-link-like handlebodies", where a handlebody $V\subset S^3$ of genus $g$ is called boundary-link-like if $π_1(\overline{S^3-V})$ admits a homomorphism to the free group $F_g$ such that the composition $π_1(\partial V)\toπ_1(\overline{S^3-V})\to F_g$ is surjective.

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arXiv2024

Two-variable Conway polynomial and Cochran's derived invariants

We note that the Conway potential function $Ω_L$ of an $m$-component link $L$, $m>1$, can be expressed as $Ω_L(x_1,\dots,x_m)=Θ_L(\nabla_L(x_1-x_1^{-1},\dots,x_m-x_m^{-1}))$ for a unique $\nabla_L\in\mathbb Z[z_1,\dots,z_m]$, where $Θ_L$ is a certain endomorphism of the additive group of $\mathbb Z[x_1^{\pm1},\dots,x_m^{\pm1}]$ which depends only on the pairwise linking numbers of the components of $L$. Motivated by applications to topological isotopy, we study the formal power series $\bar\nabla_L$, obtained by dividing $\nabla_L$ by the Conway polynomials of the components of $L$. For a $2$-component link with $lk(L)=0$, the coefficient $α_{1,2k-1}$ of $\bar\nabla_L(u,v)$ at $uv^{2k-1}$ equals Cochran's derived invariant $(-1)^{k+1}β^k(L)$. While this can be deduced from a result of G.-T. Jin, which he proved using the surgical view of the Alexander polynomial, we provide an alternative proof, using Seifert matrices. Our main result is a formula for the same coefficient $α_{1,2k-1}$ in the geometrically subtler case $lk(L)=1$. Namely we express it in terms of generalized Cochran invariants $β_F^{ij}(P,Q)$, which were studied by Gilmer--Livingston (when $P=Q$) and by Tsukamoto--Yasuhara (when $j=0$) and are closely related to the Cochran pairing in the infinite cyclic covering of a knot.

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arXiv2023

Lifting generic maps to embeddings. Triangulation and smoothing

We show that if a non-degenerate PL map $f:N\to M$ lifts to a topological embedding in $M\times\mathbb R^k$ then it lifts to a PL embedding in there. We also show that if a stable smooth map $N^n\to M^m$, $m\ge n$, lifts to a topological embedding in $M\times\mathbb R$, then it lifts to a smooth embedding in there.

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arXiv2022

Lim colim versus colim lim. I

We study a model situation in which direct limit ($\text{colim}$) and inverse limit ($\lim$) do not commute, and offer some computations of their "commutator". The homology of a separable metrizable space $X$ has two well-known approximants: $qH_n(X)$ ("Čech homology") and $pH_n(X)$ ("Čech homology with compact supports"), which are not homology theories but are nevertheless interesting as they are $\lim\text{colim}$ and $\text{colim}\lim$ applied to homology of finite simplicial complexes. The homomorphism $τ_X: pH_n(X)\to qH_n(X)$, which is a special case of the natural map $\text{colim}\lim\to\lim\text{colim}$, need not be either injective (P. S. Alexandrov, 1947) or surjective (E. F. Mishchenko, 1953), but its surjectivity for locally compact $X$ remains an open problem. In the case $n=0$ we obtain an affirmative solution of this problem. For locally compact $X$, the dual map in cohomology $pH^n(X)\to qH^n(X)$ is shown to be surjective and its kernel is computed, in terms of $\lim^1$ and a new functor $\lim^1_{\text{fg}}$. The original map $τ_X$ is surjective and its kernel is computed when $X$ is a "coronated polyhedron", i.e. contains a compactum whose complement is a polyhedron.

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arXiv2022

Fine shape I

We introduce and develop fine shape, which has a very simple definition and aims to supersede all previously known shape theories for metrizable spaces. The problem with known shape theories of metrizable spaces is illustrated by the following bizarre situation. Čech cohomology is an invariant of shape, and a fortiori of strong shape. But Steenrod-Sitnikov homology is not shape invariant (already for compacta) and has not been proved to be strong shape invariant (in more than 40 years). Worse yet, there is an ordinary homology theory that is strong shape invariant by design, but it cannot be computed in ZFC for simplest non-compact non-ANRs (such as the disjoint union of countably many copies of the one-point compactification of the countable discrete space). On the other hand, Steenrod-Sitnikov homology is an invariant of antishape (=compactly generated strong shape), and a fortiori of strong antishape. However, Čech cohomology is not antishape invariant (already for ANRs) and has not been proved to be strong antishape invariant. And there is an ordinary cohomology theory that is strong antishape invariant by design, but it cannot be computed in ZFC for simplest non-compact non-ANRs. Even though strong shape and strong antishape differ from each other by exchanging direct and inverse limits, we show that their natural "corrections" (taking into account a topology on the indexing sets) coincide for all metrizable spaces. This common "correction", called fine shape, is much simpler than the original theories and has both Čech cohomology and Steenrod-Sitnikov homology as its invariants. For ANRs fine shape coincides with homotopy, for compacta with strong shape, and for locally compact separable metrizable spaces - with strong antishape. We prove that a (co)homology theory is fine shape invariant if and only if it satisfies the map excision axiom.

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arXiv2022

Fine shape III: $Δ$-spaces and $\nabla$-spaces

In this paper we obtain results indicating that fine shape is tractable and "not too strong" even in the non-locally compact case, and can be used to better understand infinite-dimensional metrizable spaces and their homology theories. We show that every Polish space $X$ is fine shape equivalent to the limit of an inverse sequence of simplicial maps between metric simplicial complexes. A deeper result is that if $X$ is locally finite dimensional, then the simplicial maps can be chosen to be non-degenerate. They cannot be chosen to be non-degenerate if $X$ is the Taylor compactum.

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arXiv2022

Fine shape II: A Whitehead-type theorem

We prove an "abelian, locally compact" Whitehead theorem in fine shape: A fine shape morphism between locally connected finite-dimensional locally compact separable metrizable spaces with trivial $π_0$ and $π_1$ is a fine shape equivalence if and only if it induces isomorphisms on the $π_i$ (=the Steenrod-Sitnikov homotopy groups). We show by an example that the hypothesis of local connectedness cannot be dropped (even though it can be dropped in the compact case). As a byproduct, we also show that for a locally compact separable metrizable space $X$, the Steenrod-Sitnikov homology $H_n(X)=0$ if and only if each compactum $K\subset X$ lies in a compactum $L\subset X$ such that the map $H_n(K)\to H_n(L)$ is trivial. A cornerstone result of the paper is purely algebraic: If a direct sequence of groups $Γ_0\toΓ_1\to\dots$ has trivial colimit, then it is trivial as an ind-group (i.e. each $Γ_i$ maps trivially to some $Γ_j$), as long as it has one of the following forms: $\bullet$ $\lim^1_i G_{i0}\to\lim^1_i G_{i1}\to\dots$, where the $G_{ij}$ are countable abelian groups; $\bullet$ $\lim_i G_{i0}\to\lim_i G_{i1}\to\dots$, where the $G_{ij}$ are finitely generated groups, which are either all abelian or satisfy the Mittag-Leffler condition for each $j$.

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arXiv2022

Coronated polyhedra and coronated ANRs

Locally compact separable metrizable spaces are characterized among all metrizable spaces as those that admit a cofinal sequence $K_1\subset K_2\subset\cdots$ of compact subsets. Their Čech cohomology is well-understood due to Petkova's short exact sequence $0\to\lim^1 H^{n-1}(K_i)\to H^n(X)\to\lim H^n(K_i)\to 0$. We study a dual class of spaces. We call a metrizable space $X$ a "coronated polyhedron" if it contains a compactum $K$ such that $X\setminus K$ is a polyhedron. These include, apart from compacta and polyhedra, spaces such as the topologist's sine curve (or the Warsaw circle) and the comb (=comb-and-flea) space. The complement of every locally compact subset of $S^n$ is a coronated polyhedron. We prove that a metrizable space $X$ is a coronated polyhedron if and only if it admits a countable polyhedral resolution; or, equivalently, a sequential polyhedral resolution $\dots\to R_2\to R_1$. In the latter case, we establish a short exact sequence $0\to\lim^1 H_{n+1}(R_i)\to H_n(X)\to\lim H_n(R_i)\to 0$ for Steenrod-Sitnikov homology and also for any (extraordinary) homology theory satisfying Milnor's axioms of map excision and $\prod$-additivity. We also show that such homology theories are invariants of strong shape for coronated polyhedra. On the other hand, Quigley's short exact sequence $0\to\lim^1π_{n+1}(R_i)\toπ_n(X)\to\limπ_n(R_i)\to 0$ for Steenrod homotopy of compacta fails for Steenrod-Sitnikov homotopy of coronated polyhedra, at least when $n=0$.

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arXiv2022

Metrizable uniform spaces

Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spaces and A is a closed subset of X, we show that the adjunction space X\cup_f Y with the quotient uniformity (hence also with the topology thereof) is metrizable, by an explicit metric. This yields natural constructions of cone, join and mapping cylinder in the category of metrizable uniform spaces, which we show to coincide with those based on subspace (of a normed linear space); on product (with a cone); and on the isotropy of the l_2 metric. 2) We revisit Isbell's theory of uniform ANRs, as refined by Garg and Nhu in the metrizable case. The iterated loop spaces Ω^n P of a pointed compact polyhedron P are shown to be uniform ANRs. Four characterizations of uniform ANRs among metrizable uniform spaces X are given: (i) the completion of X is a uniform ANR, and the remainder is uniformly a Z-set in the completion; (ii) X is uniformly locally contractible and satisfies the Hahn approximation property; (iii) X is uniformly ε-homotopy dominated by a uniform ANR for each ε>0; (iv) X is an inverse limit of uniform ANRs with "nearly splitting" bonding maps. Several chapters are devoted primarily to exposition: (I) an introduction to uniform spaces, with a focus on the metrizable case; (V) the space of measurable functions; (VI) the space of probability measures and other measure spaces.

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arXiv2022

Infinite-dimensional uniform polyhedra

Uniform covers with a finite-dimensional nerve are rare (i.e., do not form a cofinal family) in many separable metric spaces of interest. To get hold on uniform homotopy properties of these spaces, a reasonably behaved notion of an infinite-dimensional metric polyhedron is needed; a specific list of desired properties was sketched by J. R. Isbell in a series of publications in 1959-64. In this paper we construct what appears to be the desired theory of uniform polyhedra; incidentally, considerable information about their metric and Lipschitz properties is obtained.

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arXiv2022

Lim colim versus colim lim. II: Derived limits over a pospace

Čech cohomology $H^n(X)$ of a separable metrizable space $X$ is defined in terms of cohomology of its nerves (or ANR neighborhoods) $P_β$ whereas Steenrod-Sitnikov homology $H_n(X)$ is defined in terms of homology of compact subsets $K_α\subset X$. We show that one can also go vice versa: in a sense, $H^n(X)$ can be reconstructed from $H^n(K_α)$, and if $X$ is finite dimensional, $H_n(X)$ can be reconstructed from $H_n(P_β)$. The reconstruction is via a Bousfield-Kan/Araki-Yoshimura type spectral sequence, except that the derived limits have to be "corrected" so as to take into account a natural topology on the indexing set. The corrected derived limits coincide with the usual ones when the topology is discrete, and in general are applied not to an inverse system but to a "partially ordered sheaf". The "correction" of the derived limit functors in turn involves constructing a "correct" (metrizable) topology on the order complex $|P|$ of a partially ordered metrizable space $P$ (such as the hyperspace $K(X)$ of nonempty compact subsets of $X$ with the Hausdorff metric). It turns out that three natural approaches (by using the space of measurable functions, the space of probability measures, or the usual embedding $K(X)\to C(X;\mathbb R)$) all lead to the same topology on $|P|$.

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arXiv2022

Embeddability of joins and products of polyhedra

We present a short proof of S. Parsa's theorem that there exists a compact $n$-polyhedron $P$, $n\ge 2$, non-embeddable in $\mathbb R^{2n}$, such that $P*P$ embeds in $\mathbb R^{4n+2}$. This proof can serve as a showcase for the use of geometric cohomology. We also show that a compact $n$-polyhedron $X$ embeds in $\mathbb R^m$, $m\ge\frac{3(n+1)}2$, if either - $X*K$ embeds in $\mathbb R^{m+2k}$, where $K$ is the $(k-1)$-skeleton of the $2k$-simplex; or - $X*L$ embeds in $\mathbb R^{m+2k}$, where $L$ is the join of $k$ copies of the $3$-point set; or - $X$ is acyclic and $X\times\text{(triod)}^k$ embeds in $\mathbb R^{m+2k}$.

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arXiv2022

A triple-point Whitney trick

We use a triple-point version of the Whitney trick to show that ornaments of three orientable $(2k-1)$-manifolds in $\mathbb R^{3k-1}$, $k>2$, are classified by the $μ$-invariant. A very similar (but not identical) construction was found independently by I. Mabillard and U. Wagner, who also made it work in a much more general situation and obtained impressive applications. The present note is, by contrast, focused on a minimal working case of the construction.

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