arXiv subjects

Sergey A. Melikhov

Publications and source records attributed to Sergey A. Melikhov.

41 records · Page 3Linked to original sources

arXiv2005

n-Quasi-isotopy: I. Questions of nilpotence

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 some (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C^0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-isotopy, k=1,2,...; all of them are weaker than isotopy (in the sense of Milnor). We prove that every link can be cancelled up to peripheral structure preserving isomorphism of any quotient of the fundamental group, functorially invariant under k-quasi-isotopy; functoriality means that the isomorphism between the quotients for links related by an allowable crossing change fits in the commutative diagram with the fundamental group of the complement to the intermediate singular link. The proof invokes Baer's theorem on the join of subnormal locally nilpotent subgroups. On the other hand, the integral generalized (lk\ne 0) Sato-Levine invariant \tildeβis invariant under 1-quasi-isotopy, but is not determined by any quotient of the fundamental group (endowed with the peripheral structure), functorially invariant under 1-quasi-isotopy - in contrast to Waldhausen's theorem. As a byproduct, we use \tildeβto determine the image of the Kirk-Koschorke invariant \tildeσof fibered link maps.

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arXiv2005

n-Quasi-isotopy: II. Comparison

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 (resp. contractible open 3-manifold) M if and only if M is homeomorphic to S^3 (resp. R^3). As a byproduct of the proof of the "if" part, we obtain that every compact subset of an acyclic open set in a compact orientable 3-manifold M is contained in a PL homology 3-ball in M. We show that k-quasi-isotopy implies (k+1)-cobordism of Cochran and Orr. If z^{m-1}(c_0 + c_1 z^2 + ... + c_n z^{2n}) denotes the Conway polynomial of an m-component link, it follows that the residue class of c_k modulo gcd(c_0,..,c_{k-1}) is invariant under k-quasi-isotopy. Another corollary is that each Cochran's derived invariant β^k is also invariant under k-quasi-isotopy, and therefore assumes the same value on all PL links, sufficiently C^0-close to a given topological link. This overcomes an algebraic obstacle encountered by Kojima and Yamasaki, who "became aware of impossibility to define" for wild links what for PL links is equivalent to the formal power series \sum β^n z^n by a change of variable.

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arXiv2004

Sphere eversions and realization of mappings

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 -> S^3, obtained by capping off at infinity a time-symmetric (e.g. Shapiro's) sphere eversion S^2 x I -> R^3, was non-realizable in R^6. We show that there exists a self-map of the Poincaré homology 3-sphere, non-realizable in R^6, but every self-map of S^n is realizable in R^{2n} for each n>2. The latter together with a ten-line proof for n=2, due essentially to M. Yamamoto, implies that every inverse limit of n-spheres embeds in R^{2n} for n>1, which settles R. Daverman's 1990 problem. If M is a closed orientable 3-manifold, we show that there exists a map S^3 -> M, non-realizable in R^6, if and only if π_1(M) is finite and has even order. As a byproduct, an element of the stable stem Π_3 with non-trivial stable Hopf invariant is represented by a particularly simple immersion S^3 -> R^4, namely the composition of the universal 8-covering over Q^3=S^3/{\pm1,\pm i,\pm j,\pm k}$ and an explicit embedding Q^3\subset R^4.

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arXiv2003

n-quasi-isotopy: III. Engel conditions

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 β^k, provided k>2, and a series of \barμ-invariants, starting with \barμ(111112122) for k=3, also fall in this gap. In fact, all \barμ-invariants where each index occurs at most k+1 times, except perhaps for one occuring k+2 times, can be extracted from G_k, and if they vanish, G_k is the same as that of the unlink. We also study the equivalence relation on links (called `fine k-quasi-isotopy') generated by ambient isotopy and the operation of interior connected sum with the second component of the (k+1)-th Milnor's link, where the complement to its first component is embedded into the link complement. We show that the finest quotient of the fundamental group, functorially invariant under fine k-quasi-isotopy is obtained from the fundamental group by forcing all meridians to be (k+2)-Engel elements. We prove that any group generated by two 3-Engel elements has lower central series of length <6.

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arXiv2001

On maps with unstable singularities

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 Shchepin 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)=(3,6); (1,3), and 'yes' when: 1) m>2n, (n,m)\neq (1,3); 2) 2m>3(n+1) and the set {(x,y)|f(x)=f(y)} has an equivariant (with respect to the factor exchanging involution) mapping cylinder neighborhood in X\times X; 3) m>n+2 and f is the composition of a PL map and a TOP embedding. In doing this, we answer affirmatively (with a minor preservation) a question of Kirby: does small smooth isotopy imply small smooth ambient isotopy in the metastable range, verify a conjecture of Kearton-Lickorish: small PL concordance implies small PL ambient isotopy in codimension \ge 3, and a conjecture set of Repovs-Skopenkov.

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