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Sergey A. Melikhov

Publications and source records attributed to Sergey A. Melikhov.

At least 37 records · Page 2Linked to original sources
arXiv2022

A Galois connection between classical and intuitionistic logics. I: Syntax

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 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 only new connectives ? and ! of QHC induce a Galois connection (i.e., a pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation translation of propositions into problems extends to a retraction of QHC onto QH; whereas Goedel's provability translation of problems into modal propositions extends to a retraction of QHC onto its QC+(?!) fragment, identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic modal logic, whose modality !? is a strict lax modality in the sense of Aczel - and thus resembles the squash/bracket operation in intuitionistic type theories. The axioms of QHC attempt to give a fuller formalization (with respect to the axioms of intuitionistic logic) to the two best known contentual interpretations of intiuitionistic logic: Kolmogorov's problem interpretation (incorporating standard refinements by Heyting and Kreisel) and the proof interpretation by Orlov and Heyting (as clarified by Gödel). While these two interpretations are often conflated, from the viewpoint of the axioms of QHC neither of them reduces to the other one, although they do overlap.

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arXiv2022

A Galois connection between classical and intuitionistic logics. II: Semantics

Three classes of models of QHC, the joint logic of problems and propositions, are constructed, including a class of subset/sheaf-valued models that is related to solutions of some actual problems (such as solutions of algebraic equations). To test the models, we consider a number of principles and rules, which empirically appear to cover all "sufficiently simple" natural conjectures about the behaviour of the operators ! and ?, and include two hypotheses put forward by Hilbert and Kolmogorov, as formalized in the language of QHC. Each of these turns out to be either derivable in QHC or equivalent to one of 14 principles and rules, of which 11 are conservative over classical and intuitionistic logics. The three classes of models together suffice to confirm the independence of these 11 principles and rules, and to determine all implications between them, apart from one potential implication.

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arXiv2021

The van Kampen obstruction and its relatives

We review a cochain-free treatment of the classical van Kampen obstruction θto embeddability of an n-polyhedron into R^{2n} and consider several analogues and generalizations of θ, including an extraordinary lift of θwhich in the manifold case has been studied by J.-P. Dax. The following results are obtained. - The mod 2 reduction of θis incomplete, which answers a question of Sarkaria. - An odd-dimensional analogue of θis a complete obstruction to linkless embeddability (="intrinsic unlinking") of the given n-polyhedron in R^{2n+1}. - A "blown up" 1-parameter version of θis a universal type 1 invariant of singular knots, i.e. knots in R^3 with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integer-valued type 1 invariant of singular knots admits an integral arrow diagram (=Polyak-Viro) formula. - Settling a problem of Yashchenko in the metastable range, we obtain that every PL manifold N, non-embeddable in a given R^m, m\ge 3(n+1)/2, contains a subset X such that no map N\to R^m sends X and N-X to disjoint sets. - We elaborate on McCrory's analysis of the Zeeman spectral sequence to geometrically characterize "k-co-connected and locally k-co-connected" polyhedra, which we embed in R^{2n-k} for k<(n-3)/2 extending the Penrose-Whitehead-Zeeman theorem.

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arXiv2021

Projected and near-projected embeddings

A stable smooth map $f:N\to M$ is called "$k$-realizable" if its composition with the inclusion $M\subset M\times\Bbb R^k$ is $C^0$-approximable by smooth embeddings; and a "$k$-prem" if the same composition is $C^\infty$-approximable by smooth embeddings, or equivalently if $f$ lifts vertically to a smooth embedding $N\to M\times\Bbb R^k$. It is obvious that if $f$ is a $k$-prem, then it is $k$-realizable. We refute the long-standing conjecture that the converse is always true. Namely, for each $n=4k+3\ge 15$ there exists a stable smooth immersion $S^n\to\Bbb R^{2n-7}$ that is $3$-realizable but is not a $3$-prem. We also prove the converse in a wide range of cases. A $k$-realizable stable smooth fold map $N^n\to\Bbb R^{2n-q}$ is a $k$-prem if $q\le n$ and $q\le 2k-3$; or if $q<n/2$ and $k=1$; or if $q\in\{2k-1,\,2k-2\}$ and $k\in\{2,4,8\}$ and $n$ is sufficiently large.

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arXiv2020

Topological isotopy and Cochran's derived invariants

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 $β_i$.

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arXiv2019

Self $C_2$-equivalence of two-component links and invariants of link maps

We use Kirk's invariant of link maps $S^2\sqcup S^2\to S^4$ and its variations due to Koschorke and Kirk-Livingston to deduce results about classical links. Namely, we give a new proof of the Nakanishi-Ohyama classification of two-component links in $S^3$ up to $Δ$-link homotopy. We also prove its version for string links, which is due (in a slightly different form) to Fleming-Yasuhara. The proofs do not use Clasper Theory.

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arXiv2018

Immersions of the circle into a surface

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 $e\bar f:S^1\to M_f\subset\Bbb R^2$, where $\bar f$ is a lift of $f$ to the cover $M_f$ of $M$ corresponding to the subgroup $\left<[f]\right>\subsetπ_1(M)$. Namely, immersions $f,g:S^1\to M$ are regular homotopic if and only if they are homotopic, and if $M=S^2$ or $\Bbb R P^2$ or the normal bundle $ν(f)$ is non-orientable, then $S(f)=S(g)$, whereas if $M\not= S^2,\Bbb R P^2$ and $ν(f)$, $ν(g)$ have orientations $o$, $o'$, compatible with respect to the homotopy, then $T(e_o\bar f)=T(e_{o'}\bar g)$, where $e_o$ is a standard embedding of the oriented surface $M_f$ (an annulus or a plane) in $\Bbb R^2$. In fact, for homotopic immersions $f$, $g$ both $S(f)-S(g)$ and $T(e_o\bar f)-T(e_{o'}\bar g)$ boil down to the turning number of a lift of a null-homotopic immersion $f\# g^*$ to the universal cover of $M$. Here "immersions" $S^1\to M$ are either smooth or topological; we include a smoothing theorem, which shows that there is no difference. We also classify immersions of a graph in $M$ up to regular homotopy in terms of the invariants $S(f)$ and $T(e_o\bar f)$ of immersed $S^1$'s. The proofs are based on the h-principle. The point of this unsophisticated note is to simplify [10] and [11], where a classification of immersions of a graph in $M$ was obtained for $M\ne\Bbb R P^2$ in terms of a rather laboriously defined "winding number" of a pair of homotopic immersions $S^1\to M$ (rather than of an individual immersion) with respect to a given vector field with zeroes on $M$.

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arXiv2018

Combinatorics of combinatorial topology

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 of constructible posets (generalizing face posets of constructible simplicial complexes in the sense of Hochster). The relevant constructible posets are also characterized in terms of Reading's zipping of posets, which is a variation of edge contractions in simplicial complexes.

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arXiv2018

Algebraic topology of Polish spaces. II: Axiomatic homology

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 additivity axioms suffice to uniquely characterize (co)homology of closed pairs of Polish spaces (=separable complete metrizable spaces). The proof provides a combinatorial description of the (co)homology of a Polish space in terms of a cellular (co)chain complex satisfying a symmetry of the form $\lim \text{colim} = \text{colim} \lim$.

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arXiv2017

Gauss-type formulas for link map invariants

We find that Koschorke's $β$-invariant and the triple $μ$-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 were unexpected for the author, because the only known direct ways to extract $μ$ and $β$ from invariants of maps between configuration spaces involved some homotopy theory (Whitehead products and the stable Hopf invariant, respectively).

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arXiv2017

Mathematical semantics of intuitionistic logic

This work is a mathematician's attempt to understand intuitionistic logic. It can be read in two ways: as a research paper interspersed with lengthy digressions into rethinking of standard material; or as an elementary (but highly unconventional) introduction to first-order intuitionistic logic. For the latter purpose, no training in formal logic is required, but a modest literacy in mathematics, such as topological spaces and posets, is assumed. The main theme of this work is the search for a formal semantics adequate to Kolmogorov's informal interpretation of intuitionistic logic (whose simplest part is more or less the same as the so-called BHK interpretation). This search goes beyond the usual model theory, based on Tarski's notion of semantic consequence, and beyond the usual formalism of first-order logic, based on schemata. Thus we study formal semantics of a simplified version of Paulson's meta-logic, used in the Isabelle prover. By interpreting the meta-logical connectives and quantifiers constructively, we get a generalized model theory, which covers, in particular, realizability-type interpretations of intuitionistic logic. On the other hand, by analyzing Kolmogorov's notion of semantic consequence (which is an alternative to Tarski's standard notion), we get an alternative model theory. By using an extension of the meta-logic, we further get a generalized alternative model theory, which suffices to formalize Kolmogorov's semantics. On the other hand, we also formulate a modification of Kolmogorov's interpretation, which is compatible with the usual, Tarski-style model theory. Namely, it can be formalized by means of sheaf-valued models, which turn out to be a special case of Palmgren's categorical models; intuitionistic logic is complete with respect to this semantics.

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arXiv2016

Transverse fundamental group and projected embeddings

For a generic degree d smooth map f: N^n -> M^n we introduce its "transverse fundamental group" π(f), which reduces to π_1(M) in the case where f is a covering, and in general admits a monodromy homomorphism π(f) -> S_{|d|}; nevertheless, we show that π(f) can be non-trivial already for rather simple degree 1 maps S^n -> S^n. We apply π(f) to the problem of lifting f to an embedding N -> M x R^2: for such a lift to exist, the monodromy π(f) -> S_{|d|} must factor through the group of concordance classes of |d|-component string links. At least if |d|<7, this requires π(f) to be torsion-free.

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arXiv2012

Contractible polyhedra in products of trees and absolute retracts in products of dendrites

We show that a compact n-polyhedron PL embeds in a product of n trees if and only if it collapses onto an (n-1)-polyhedron. If the n-polyhedron is contractible and n\ne 3 (or 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, there exists a 2-dimensional compact absolute retract X such that X\times I^k does not embed in any product of 2+k dendrites for each k.

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arXiv2011

Combinatorics of embeddings

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 is PL homeomorphic to a sphere; there exist precisely two 3-dimensional dichotomial cell complexes, and their 1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their 1-skeleta all but one graphs of the Petersen family. In higher dimensions n>2, we observe that in order to characterize those compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited minors", it suffices to establish finiteness of the list of all (n-1)-connected n-dimensional finite cell complexes that do not embed in S^{2n} yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the n-skeleta of (2n+1)-dimensional dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell complexes include (apart from the three joins of the i-skeleta of (2i+2)-simplices) at least ten non-simplicial complexes.

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arXiv2009

Steenrod homotopy

Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homotopy coincides with strong shape. We attempt to simplify foundations of the theory and to clarify and improve some of its major results. Using geometric tools such as Milnor's telescope compactification, comanifolds (=mock bundles) and the Pontryagin-Thom Construction, we obtain new simple proofs of results by Barratt-Milnor; Cathey; Dydak-Segal; Eda-Kawamura; Edwards-Geoghegan; Fox; Geoghegan-Krasinkiewicz; Jussila; Krasinkiewicz-Minc; Mardesic; Mittag-Leffler/Bourbaki; and of three unpublished results by Shchepin. An error in Lisitsa's proof of the "Hurewicz theorem in Steenrod homotopy" is corrected. It is shown that over compacta, R.H.Fox's overlayings are same as I.M.James' uniform covering maps. Other results include: - A morphism between inverse sequences of countable (possibly non-abelian) groups that induces isomorphisms on inverse and derived limits is invertible in the pro-category. This implies the "Whitehead theorem in Steenrod homotopy", thereby answering two questions of A.Koyama. - If X is an LC_{n-1} compactum, n>0, its n-dimensional Steenrod homotopy classes are representable by maps S^n\to X, provided that X is simply connected. The assumption of simply-connectedness cannot be dropped by a well-known example of Dydak and Zdravkovska. - A connected compactum is Steenrod connected (=pointed 1-movable) iff every its uniform covering space has countably many uniform connected components.

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arXiv2006

The telescope approach to embeddability of compacta

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 - Δadmits 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 - Δis trivial. Assuming that X quasi-embeds in R^{2n} (i.e. is an inverse limit of n-polyhedra, embeddable in R^{2n}), this is equivalent to the vanishing of an obstruction in lim^1 H^{2n-1}(K_i) over compact subsets K_i\subset X x X - Δ. One application is that an n-dimensional ANR embeds in R^{2n} if it quasi-embeds in R^{2n-1}, n>3. We construct an ANR of dimension n>1, quasi-embeddable but not embeddable in R^{2n}, and an AR of dimension n>1, which does not "movably" embed in R^{2n}. These examples come close to, but don't quite resolve, Borsuk's problem: does every n-dimensional AR embed in R^{2n}? In the affirmative direction, we show that an n-dimensional compactum X embeds in R^{2n}, n>3, if H^n(X)=0 and H^{n+1}(X,X-pt)=0 for every pt\in X. There are applications in the entire metastable range as well. An n-dimensional compactum X with H^{n-i}(X-pt)=0 for each pt\in X and all i\le k embeds in R^{2n-k}. This generalizes Bryant and Mio's result that k-connected n-dimensional generalized manifolds embed in R^{2n-k}. Also, an acyclic compactum X embeds in R^m iff X x I embeds in R^{m+1} iff X x (triod) embeds in R^{m+2}. As a byproduct, we answer a question of T. Banakh on stable embeddability of the Menger cube.

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