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Dusan Repovs

Publications and source records attributed to Dusan Repovs.

14 recordsLinked to original sources
arXiv2014

Sequential rectifiable spaces of countable $cs^*$-character

We prove that each non-metrizable sequential rectifiable space $X$ of countable $cs^*$-character contains a clopen rectifiable submetrizable $k_\omega$-subspace $H$ and admits an open disjoint cover by subspaces homeomorphic to clopen subspaces of $H$. This implies that each sequential rectifiable space with countable $cs^*$-character either is metrizable or else is a topological sum of submetrizable $k_\omega$-spaces. Consequently, $X$ is submetrizable and paracompact. This answers a question of Lin and Shen posed in 2011.

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arXiv2014

Classifying homogeneous cellular ordinal balleans up to coarse equivalence

For every ballean $X$ we introduce two cardinal characteristics $cov^\flat(X)$ and $cov^\sharp(X)$ describing the capacity of balls in $X$. We observe that these cardinal characteristics are invariant under coarse equivalence and prove that two cellular ordinal balleans $X,Y$ are coarsely equivalent if $cof(X)=cof(Y)$ and $cov^\flat(X)=cov^\sharp(X)=cov^\flat(Y)=cov^\sharp(Y)$. This result implies that a cellular ordinal ballean $X$ is homogeneous if and only if $cov^\flat(X)=cov^\sharp(X)$. Moreover, two homogeneous cellular ordinal balleans $X,Y$ are coarsely equivalent if and only if $cof(X)=cof(Y)$ and $cov^\sharp(X)=cov^\sharp(Y)$ if and only if each of these balleans coarsely embeds into the other ballean. This means that the coarse structure of a homogeneous cellular ordinal ballean $X$ is fully determined by the values of the cardinals $cof(X)$ and $cov^\sharp(X)$. For every limit ordinal $\gamma$ we shall define a ballean $2^{<\gamma}$ (called the Cantor macro-cube), which in the class of cellular ordinal balleans of cofinality $cf(\gamma)$ plays a role analogous to the role of the Cantor cube $2^{\kappa}$ in the class of zero-dimensional compact Hausdorff spaces. We shall also present a characterization of balleans which are coarsely equivalent to $2^{<\gamma}$. This characterization can be considered as an asymptotic analogue of Brouwer's characterization of the Cantor cube $2^\omega$.

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arXiv2013

Universal meager $F_\sigma$-sets in locally compact manifolds

In each manifold $M$ modeled on a finite or infinite dimensional cube $[0,1]^n$ we construct a meager $F_\sigma$-subset $X\subset M$ which is universal meager in the sense that for each meager subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h(A)\subset X$. We also prove that any two universal meager $F_\sigma$-sets in $M$ are ambiently homeomorphic.

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arXiv2013

Universal nowhere dense and meager sets in Menger manifolds

In each Menger manifold $M$ we construct: (i) a closed nowhere dense subset $M_0$ which is homeomorphic to $M$ and is universal nowhere dense in the sense that for each nowhere dense set $A\subset M$ there is a homeomorphism $h$ of $M$ such that $h(A)\subset M_0$; (ii) a meager $F_\sigma$-set $\Sigma_0\subset M$ which is universal meager in the sense that for each meager subset $B\subset M$ there is a homeomorphism $h$ of $M$ such that $h(B)\subset \Sigma_0$. Also we prove that any two universal meager $F_\sigma$-sets in $M$ are ambiently homeomorphic.

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arXiv2013

Universal nowhere dense subsets of locally compact manifolds

In each manifold $M$ modeled on a finite or infinite dimensional cube $[0,1]^n$ we construct a closed nowhere dense subset $S\subset M$ (called a spongy set) which is a universal nowhere dense set in $M$ in the sense that for each nowhere dense subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h(A)\subset S$. The key tool in the construction of spongy sets is a theorem on topological equivalence of certain decompositions of manifolds. A special case of this theorem says that two vanishing cellular strongly shrinkable decompositions $\mathcal A,\mathcal B$ of a Hilbert cube manifold $M$ are topologically equivalent if any two non-singleton elements $A\in\mathcal A$ and $B\in\mathcal B$ of these decompositions are ambiently homeomorphic.

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arXiv2010

A new Lindelof topological group

We show that the subsemigroup of the product of w_1-many circles generated by the L-space constructed by J. Moore is again an L-space. This leads to a new example of a Lindelof topological group. The question whether all finite powers of this group are Lindelof remains open.

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arXiv2009

Direct limit topologies in the categories of topological groups and of uniform spaces

We study the topological structure of the direct limit $\glim G_n$ of a tower of topological groups $(G_n)$ in the category of topological groups and show that under some conditions on the tower $(G_n)$ the topology of $\glim G_n$ coincides with the topology of the direct limit $\ulim G_n$ of the groups $G_n$ endowed with the Roelcke uniformity in the category of uniform spaces.

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arXiv2009

The topological structure of direct limits in the category of uniform spaces

Let $(X_n)_{n}$ be a sequence of uniform spaces such that each space $X_n$ is a closed subspace in $X_{n+1}$. We give an explicit description of the topology and uniformity of the direct limit $u-lim X_n$ of the sequence $(X_n)$ in the category of uniform spaces. This description implies that a function $f:u-lim X_n\to Y$ to a uniform space $Y$ is continuous if for every $n$ the restriction $f|X_n$ is continuous and regular at the subset $X_{n-1}$ in the sense that for any entourages $U\in\U_Y$ and $V\in\U_X$ there is an entourage $V\in\U_X$ such that for each point $x\in B(X_{n-1},V)$ there is a point $x'\in X_{n-1}$ with $(x,x')\in V$ and $(f(x),f(x'))\in U$. Also we shall compare topologies of direct limits in various categories.

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arXiv2007

Groups which are not properly 3-realizable

A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it has {\em pro-(finitely generated free) fundamental group at infinity} and {\em semi-stable ends}. Conjecturally the quasi-simply filtration assumption is superfluous. Using these restrictions we provide the first examples of finitely presented groups which are not properly 3-realizable, for instance large families of Coxeter groups.

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arXiv2005

On topological properties of the Hartman--Mycielski functor

We investigate some topological properties of a normal functor $H$ introduced earlier by Radul which is some functorial compactification of the Hartman--Mycielski construction HM. We prove that the pair ($HX$, HM$Y$) is homeomorphic to the pair $(Q,σ)$ for each nondegenerated metrizable compactum $X$ and each dense $σ$-compact subset $Y$.

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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.

math.GT↗
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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