# Sergey MelikhovRussian Academy of Sciences | RAS

Sergey Melikhov

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49

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Introduction

## Publications

Publications (49)

A stable smooth map f : N → M is called k-realizable if its composition with the inclusion M ⊂ M × ℝk is C⁰-approximable by smooth embeddings; and a k-prem if the same composition is C∞-approximable by smooth embeddings, or, equivalently, if f lifts vertically to a smooth embedding N ↪ M × ℝk.
It is obvious that if f is a k-prem, then it is k-reali...

We construct a link in the 3 3 -space that is not isotopic to any PL link (non-ambiently). In fact, we show that there exist uncountably many I I -equivalence classes of links.
The paper also includes some observations on Cochran’s invariants β i \beta _i .

We construct a link in the $3$-space that is not isotopic to any PL link (non-ambiently). In fact, there exist uncountably many $I$-equivalence classes of links. The paper also includes some observations on Cochran's invariants $\beta_i$.

We note a simple algebraic proof of Frolkina’s result that ℝ^3 does not contain uncountably many pairwise disjoint copies of the Möbius band, and of a similar result in higher dimensions. Full text: https://arxiv.org/abs/1810.04089

We use a triple-point version of the Whitney trick to show that ornaments of three orientable [Formula: see text]-manifolds in [Formula: see text], [Formula: see text], are classified by the [Formula: see text]-invariant.
A very similar (but not identical) construction was found independently by I. Mabillard and U. Wagner, who also made it work in...

We prove the analogue of the Concordance Implies Isotopy in Codimension ≥3 Theorem for link maps, together with some other its singular analogues. In the case of spherical link maps, a stronger result was independently obtained by P. Teichner (by different methods).
Full text: https://arxiv.org/abs/1810.08299

We note a simple algebraic proof of Frolkina's result that R^3 does not contain uncountably many pairwise disjoint copies of the Möbius band, and of a similar result in higher dimensions.
Full text: https://arxiv.org/abs/1810.04089

Milnor proved two uniqueness theorems for axiomatic (co)homology: one for pairs of compacta (1960) and another, in particular, for pairs of countable simplicial complexes (1961). We obtain their common generalization: the Eilenberg-Steenrod axioms along with Milnor's map excision axiom and a (non-obvious) common generalization of Milnor's two addit...

Fine shape of Polish spaces is a common "correction" of strong shape and compactly generated strong shape (which differ from each other essentially by permuting a direct limit with an inverse limit). For compacta fine shape coincides with strong shape, and in general it involves a stronger form of coherence which amounts to taking into account a na...

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⊂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 di...

We study a model situation in which direct limit (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 ne...

We classify immersions f of S^1 in a 2-manifold M in terms of elementary invariants: the parity S(f) of the number of double points of a self-transverse C^1-approximation of f, and the turning number T(e\bar f) of the immersion e\bar f:S^1→M_f⊂R^2, where \bar f is a lift of f to the cover M_f of M corresponding to the subgroup ⟨[f]⟩⊂π_1(M). Namely,...

Погружения f окружности в двумерное многообразие M расклассифицированы в терминах элементарных инвариантов: четности S(f) числа двойных точек самотрансверсальной C^1-аппроксимации f и числа вращения T(e\bar f) погружения e\bar f:S^1→M_f⊂R^2, где \bar f – поднятие f в накрытие M_f поверхности M, соответствующее подгруппе ⟨[f]⟩⊂π_1(M).
А именно, пог...

Given a generic PL map or a generic smooth immersion $f:N^n\to M^m$, where $m\ge n$ and $2(m+k)\ge 3(n+1)$, we prove that $f$ lifts to a PL or smooth embedding $N\hookrightarrow M\times\mathbb R^k$ if and only if its double point locus $(f\times f)^{-1}(\Delta_M)\setminus\Delta_N$ admits an equivariant map to $S^{k-1}$. As a corollary we answer a 1...

We use Kirk's invariant of link maps $S^2\sqcup S^2\to S^4$ and its variations to give a new simple proof of the Nakanishi-Ohyama classification of $2$-component links in $S^3$ up to $\Delta$-link homotopy, and to obtain its version for string links.
Full text: https://arxiv.org/abs/1711.03514

We find that Koschorke's $\beta$-invariant and the triple $\mu$-invariant of link maps in the critical dimension can be computed as degrees of certain maps of configuration spaces - just like the linking number. Both formulas admit geometric interpretations in terms of Vassiliev's ornaments via new operations akin to the Jin suspension, and both we...

We call a map $N\to M$ "$k$-realizable" if its composition with the inclusion $M\subset M\times\mathbb R^k$ is $C^0$-approximable by embeddings, and a "$k$-prem" if it is the vertical projection of some embedding $N\hookrightarrow M\times\mathbb R^k$. Our results include: (i) A generic PL map or a generic smooth immersion $N^n\to\mathbb R^m$ is $k$...

We obtain a classification of immersions $f$ of $S^1$ in a $2$-manifold $M$ in terms of simple geometric invariants: the parity $S(f)$ of the number of double points of $f$ and the turning number $T(f,o)$ of $\phi_o\bar f$, where $\bar f$ is a lift of $f$ to the cover $M_f$ of $M$ corresponding to $\left<[f]\right>\subset\pi_1(M)$ and $\phi_o:M_f\t...

For a generic degree d smooth map f: N^n -> M^n we introduce its "transverse fundamental group" \pi(f), which reduces to \pi_1(M) in the case where f is a covering, and in general admits a monodromy homomorphism \pi(f) -> S_{|d|}; nevertheless, we show that \pi(f) can be non-trivial already for rather simple degree 1 maps S^n -> S^n. We apply \pi(f...

This is an elementary introduction to intuitionistic logic, assuming a modest literacy in mathematics (such as topological spaces and posets) but no training in formal logic. We adopt and develop Kolmogorov's understanding of intuitionistic logic as the logic of schemes of solutions of mathematical problems. Here intuitionistic logic is viewed as a...

Abstract (part I): In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "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 object...

We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a number of known results.
New results include a combinatorial characterization of collapsible polyhedra in terms...

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 i...

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...

We offer the following explanation of the statement of the Kuratowski graph
planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas
intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to
every cell there corresponds a unique cell with the complementary set of
vertices. Then every dichotomial cell complex...

We show that a compact n n -polyhedron PL embeds in a product of n n trees if and only if it collapses onto an ( n − 1 ) (n-1) -polyhedron. If the n n -polyhedron is contractible and n ≠ 3 n\ne 3 (or n = 3 n=3 and the Andrews–Curtis Conjecture holds), the product of trees may be assumed to collapse onto the image of the embedding.
In contrast, ther...

We show that a collapsible n-polyhedron embeds in a product of n trees;
consequently, a contractible compact n-polyhedron cross I^k embeds in a product
of n+k trees for some k (by known examples k=0 does not suffice). On the other
hand, a certain 2-dimensional compact absolute retract cross I^k does not embed
in a product of 2+k dendrites (=one-dim...

Steenrod homotopy theory is a natural framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; or from a different viewpoint, it studies the topology of the functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homotopy coincides with st...

A continuous map of a compact -polyhedron into an orientable piecewise linear -manifold, , is discretely (isotopically) realizable if it is the uniform limit of a sequence of embeddings , (respectively, of an isotopy , ), and is continuously realizable if any embedding sufficiently close to can be included in an arbitrarily small such isotopy. It w...

We show that an n-dimensional compactum X embeds in R^m, where m>3(n+1)/2, if and only if X x X - \Delta admits an equivariant map to S^{m-1}. In particular, X embeds in R^{2n}, n>3, iff the top power of the (twisted) Euler class of the factor-exchanging involution on X x X - \Delta is trivial. Assuming that X quasi-embeds in R^{2n} (i.e. is an inv...

We review a cochain-free treatment of the classical van Kampen obstruction ϑ to embeddability of an n-polyhedron in ℝ2n
and consider several analogs and generalizations of ϑ, including an extraordinary lift of ϑ, which has been studied by J.-P. Dax in the manifold case. The following results are obtained: (1) The mod 2 reduction of ϑ is incomplete,...

Geometric aspects of the filtration on classical links by k-quasi-isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: ω-quasi-isotopy is equivalent to PL isotopy for links in a homotopy 3-sphere (respectively, contractible open 3-manifold) M if...

In this paper we study the isotopic realization problem, which is the question of isotopic realizability, of a given (continuous) map f, that is, the possibility of a uniform approximation of f by a continuous family of embeddings g(t), t G [0, 00), under the condition that f is discretely realizable, that is, that there exists a uniform approximat...

The multi-variable Alexander polynomial (in the form of Conway's potential function), when stripped of a redundant summand, is shown to be of the form \Nabla_L(x_1-x_1^{-1},..,x_m-x_m^{-1}) for some polynomial \Nabla_L over Z. The Conway polynomial \nabla_L(z) coincides with z\Nabla_L(z,..,z). The coefficients of \Nabla_L and of the power series \N...

P. M. Akhmetiev used a controlled version of the stable Hopf invariant to show that any (continuous) map N -> M between stably parallelizable compact n-manifolds, n\ne 1,2,3,7, is realizable in R^{2n}, i.e. the composition of f with an embedding M\subset R^{2n} is C^0-approximable by embeddings. It has been long believed that any degree 2 map S^3 -...

In the metastable range, a class of mappings yielding a negative solution of the isotopic realization problem (posed by E. V. Shchepin in 1993) and satisfying an additional technical condition is described in algebraic terms. Namely, one constructs an obstruction to isotopic realization of a discretely realizable continuous mapping f of an n-polyhe...

If a continuous map f:X→Q is approximable arbitrary closely by embeddings X↪Q, can some embedding be taken onto f by a pseudo-isotopy? This question, called Isotopic Realization Problem, was raised by Ščepin and Akhmet'ev. We consider the case where X is a compact n-polyhedron, Q a PL m-manifold and show that the answer is ‘generally no’ for (n,m)=...

In part I it was shown that for each k>0 the generalized Sato-Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient G_k of its fundamental group, `functorially' invariant under k-quasi-isotopy. Here we show that Cochran's derived invariant \beta^k, provided k>2, and a s...

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It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for (piecewise-linear) links up to some stronger analogue of link...

Geometric aspects of the filtration on classical links by k-quasi-isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: \omega-quasi-isotopy is equivalent to PL isotopy for links in a homotopy 3-sphere (resp. contractible open 3-manifold) M if an...