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D. Repovš

Publications and source records attributed to D. Repovš.

At least 19 recordsLinked to original sources
arXiv2022

Qualitative properties of space-dependent SIR models with constant delay and their numerical solutions

In this article a space-dependent epidemic model equipped with a constant latency period is examined. We construct a delay partial integro-differential equation and show that its solution possesses some biologically reasonable features. We propose some numerical schemes and show that by choosing the time step to be sufficiently small the schemes preserve the qualitative properties of the original continuous model. Finally, some numerical experiments are presented that confirm the aforementioned theoretical results.

math.NA↗
arXiv2021

On the Browder-Levine-Novikov embedding theorems

In this survey we present applications of the ideas of complement and neighborhood in the theory embeddings of manifolds into Euclidean space (in codimension at least three). We describe how the combination of these ideas gives a reduction of embeddability and isotopy problems to algebraic problems. We present a more clarified exposition of the Browder-Levine theorem on realization of normal systems. Most of the survey is accessible to non-specialists in the theory of embeddings.

math.GT↗
arXiv2016

A four-dimensional simple algebra with fractional PI-exponent

Numerical characteristics of identities of finite-dimensional nonassociative algebras are studied. The main result is the construction of a four-dimensional simple unitary algebra with fractional PI-exponent strictly less than its dimension.

math.RA↗
arXiv2012

Classification of knotted tori in the 2-metastable dimension

This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings $N\to S^m$. We study the specific case of knotted tori, i. e. the embeddings $S^p \times S^q \to S^m$. The classification of knotted tori up to isotopy in the metastable dimension range $m>p+\frac{3}{2}q+2$, $p\le q$, was given by A. Haefliger, E. Zeeman and A. Skopenkov. We consider the dimensions below the metastable range, and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension: Theorem. Assume that $p+\frac{4}{3}q+2 2p+q+2$. Then the set of smooth embeddings $S^p \times S^q \to S^m$ up to isotopy is infinite if and only if either $q+1$ or $p+q+1$ is divisible by 4. Our approach to the classification is based on an analogue of the Koschorke exact sequence from the theory of link maps. This sequence involves a new $β$-invariant of knotted tori. The exactness is proved using embedded surgery and the Habegger-Kaiser techniques of studying the complement.

math.GT↗
arXiv2012

On manifolds with nonhomogeneous factors

We present simple examples of finite-dimensional connected homogeneous spaces (they are actually topological manifolds) with nonhomogeneous and nonrigid factors. In particular, we give an elementary solution of an old problem in general topology concerning homogeneous spaces.

math.GT↗
arXiv2011

Fuzzy Prokhorov metric on the set of probability measures

We introduce a fuzzy metric on the set of probability measures on a fuzzy metric space. The construction is an analogue, in the realm of fuzzy metric spaces, of the Prokhorov metric on the set of probability measures on compact metric spaces.

math.GN↗
arXiv2011

Locally $G$-homogeneous Busemann $G$-spaces

We present short proofs of all known topological properties of general Busemann $G$-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally $G$-homogeneous Busemann $G$-spaces are homeomorphic and strongly topologically homogeneous. This is a key result in the context of the classical Busemann conjecture concerning the characterization of topological manifolds, which asserts that every $n$-dimensional Busemann $G$-space is a topological $n$-manifold. We also prove that every Busemann $G$-space which is uniformly locally $G$-homogeneous on an orbal subset must be finite-dimensional.

math.GT↗
arXiv2009

One-relator groups and proper 3-realizability

How different is the universal cover of a given finite 2-complex from a 3-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group $G$ is said to be properly 3-realizable if there exists a compact 2-polyhedron $K$ with $π_1(K) \cong G$ whose universal cover $\tilde{K}$ has the proper homotopy type of a PL 3-manifold (with boundary). In this paper, we study the asymptotic behavior of finitely generated one-relator groups and show that those having finitely many ends are properly 3-realizable, by describing what the fundamental pro-group looks like, showing a property of one-relator groups which is stronger than the QSF property of Brick (from the proper homotopy viewpoint) and giving an alternative proof of the fact that one-relator groups are semistable at infinity.

math.GT↗
arXiv2008

Constructing near-embeddings of codimension one manifolds with countable dense singular sets

The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an embedding, i.e. $f$ is a {\sl near-embedding}.

math.GT↗
arXiv2008

Homotopy type of the complement of an immersion and classification of embeddings of tori

This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\times S^q\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high dimension, namely in the metastable range $m\ge p+3q/2+2$, $p\le q$, which is a natural limit for the classical methods of embedding theory, has been explicitely described earlier. The aim of this note is to present an approach which allows for results in lower dimension.

math.GT↗
arXiv2008

Preserving $Z$-sets by Dranishnikov's resolution

We prove that Dranishnikov's $k$-dimensional resolution $d_k\colon μ^k\to Q$ is a UV$^{n-1}$-divider of Chigogidze's $k$-dimensional resolution $c_k$. This fact implies that $d_k^{-1}$ preserves $Z$-sets. A further development of the concept of UV$^{n-1}$-dividers permits us to find sufficient conditions for $d_k^{-1}(A)$ to be homeomorphic to the Nöbeling space $ν^k$ or the universal pseudoboundary $σ^k$. We also obtain some other applications.

math.GN↗
arXiv2007

On small homotopies of loops

Two natural questions are answered in the negative: (1) If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic? (2) Can adding arcs to a space cause an essential curve to become nulhomotopic? The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and $π_1$-shape injective.

math.GT↗
arXiv2007

Classification of framed links in 3-manifolds

We present a short proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in details: {\bf Theorem.} Let $M$ be a connected oriented closed smooth 3-manifold. Let $L_1(M)$ be the set of framed links in $M$ up to a framed cobordism. Let $°:L_1(M)\to H_1(M;\Z)$ be the map taking a framed link to its homology class. Then for each $α\in H_1(M;\Z)$ there is a 1-1 correspondence between the set $°\nolimits^{-1}α$ and the group $\Bbb Z_{2d(α)}$, where $d(α)$ is the divisibility of the projection of $α$ to the free part of $H_1(M;\Bbb Z)$.

math.GT↗
arXiv2007

On $π- π$ theorem for manifold pairs with boundaries

Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(π_1(Y)\toπ_1(X))$. A well known result of Wall, the so called $π$-$π$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with $π_1(X)\congπ_1(Y)$ is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced surgery obstruction group for manifold pairs $LP_*$ and splitting obstruction groups $LS_*$. In the present paper we formulate and prove for manifold pairs with boundaries the results which are similar to the $π$-$π$ theorem. We give direct geometric proofs, which are based on the original statements of Wall's results and apply obtained results to investigate surgery on filtered manifolds.

math.GT↗
arXiv2006

Amalgamated products and properly 3-realizable groups

In this paper, we show that the class of all properly 3-realizable groups is closed under amalgamated free products (and HNN-extensions) over finite groups. We recall that $G$ is said to be properly 3-realizable if there exists a compact 2-polyhedron $K$ with $π_1(K) \cong G$ and whose universal cover $\tilde{K}$ has the proper homotopy type of a 3-manifold (with boundary).

math.GT↗
arXiv2006

Surgery spectral sequence and stratified manifolds

Cappell and Shaneson pointed out in 1978 interesting properties of Browder - Livesay invariants which are similar to differentials in some spectral sequence. Such spectral sequence was constructed in 1991 by Hambleton and Kharshiladze. This spectral sequence is closely related to a problem of realization of elements of Wall groups by normal maps of closed manifolds. The main step of construction of the spectral sequence is an infinite filtration of spectra in which only the first two, as is well-known, have clear geometric sense. The first one is a spectrum $\Bbb L(π_1(X))$ for surgery obstruction groups of a manifold $X$ and the second $\Bbb LP_*(F)$ is a spectrum for surgery on a Browder-Livesay manifold pair $Y\subset X$. The geometric sense of the third term of filtration was explained by Muranov, Repovš, and Spaggiari in 2002. In the present paper we give a geometric interpretation of all spectra of filtration in construction of Hambleton and Kharshiladze. We introduce groups of obstructions to surgery on a system of embeddedd manifolds and prove that spectra which realize these groups coincide with spectra in the filtration of Hambleton and Kharshiladze. We describe algebraic and geometric properties of introduced obstruction groups and their relations to the classical surgery theory. We prove isomorphism between introduced groups and Browder-Quinn $L$-groups of stratified manifolds. We give an application of our results to closed manifold surgery problem and iterated Browder-Livesay invariant.

math.GT↗