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In this paper first we construct the Eilenberg-MacLane space through Moore space then we study this space through an infinite symmetric product functor. Finally, we study some applications. [ABSTRACT FROM AUTHOR]

The topic of the paper is the investigation of the homology groups of the $(2n+1)$-dimensional CW-complex $\mathbb{C}\Omega_n$. The spaces $\mathbb{C}\Omega_n$ consist of complex-valued functions and are the analogue of the spaces $\Omega_n$, widely known in the approximation theory. The spaces $\mathbb{C}\Omega_n$ have been introduced in 2015 by A.M. Pasko who has built the CW-structure of the spaces $\mathbb{C}\Omega_n$ and using this CW-structure established that the spaces $\mathbb{C}\Omega_n$ are simply connected. Note that the mentioned CW-structure of the spaces $\mathbb{C}\Omega_n$ is the analogue of the CW-structure of the spaces $\Omega_n$ constructed by V.I. Ruban. Further A.M. Pasko found the homology groups of the space $\mathbb{C}\Omega_n$ in the dimensionalities $0, 1, \ldots, n, 2n-1, 2n, 2n+1$. The goal of the present paper is to find the homology group $H_{n+1}\left ( \mathbb{C}\Omega_n \right )$. It is proved that $H_{n+1} \left ( \mathbb{C}\Omega_n \right )=\mathbb{Z}^\frac{n+1}{2}$ if $n$ is odd and $H_{n+1} \left ( \mathbb{C}\Omega_n \right )=\mathbb{Z}^\frac{n+2}{2}$ if $n$ is even.

In the present paper the spaces $\Omega_n(m)$ are considered. The spaces $\Omega_n(m)$, introduced in 2018 by A.M. Pasko and Y.O. Orekhova, are the generalization of the spaces $\Omega_n$ (the space $\Omega_n(2)$ coincides with $\Omega_n$). The investigation of homotopy properties of the spaces $\Omega_n$ has been started by V.I. Ruban in 1985 and followed by V.A. Koshcheev, A.M. Pasko. In particular V.A. Koshcheev has proved that the spaces $\Omega_n$ are simply connected. We generalized this result proving that all the spaces $\Omega_n(m)$ are simply connected. In order to prove the simply connectedness of the space $\Omega_n(m)$ we consider the 1-skeleton of this space. Using 1-cells we form the closed ways that create the fundamental group of the space $\Omega_n(m)$. Using 2-cells we show that all these closed ways are equivalent to the trivial way. So the fundamental group of the space $\Omega_n(m)$ is trivial and the space $\Omega_n(m)$ is simply connected.

In the paper the homology groups of the $(2n+1)$-dimensional CW-complex $\mathbb{C}\Omega_n$ are investigated. The spaces $\mathbb{C}\Omega_n$ consist of complex-valued functions and generalize the widely known in the approximation theory spaces $\Omega_n$. The research of the homotopy properties of the spaces $\Omega_n$ has been started by V.I. Ruban who in 1985 found the n-dimensional homology group of the space $\Omega_n$ and in 1999 found all the cohomology groups of this space. The spaces $\mathbb{C}\Omega_n$ have been introduced by A.M. Pasko who in 2015 has built the structure of CW-complex on these spaces. This CW-structure is analogue of the CW-structure of the space $\Omega_n$ introduced by V.I. Ruban. In present paper in order to investigate the homology groups of the spaces $\mathbb{C}\Omega_n$ we calculate the relative homology groups $H_k(\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1})$, it turned out that the groups $H_k \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )$ are trivial if $1\leq k < n$ and $H_k \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )=\mathbb{Z}^{C^{k-n}_{n+1}}$ if $n \leq k \leq 2n+1$, in particular $H_n \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )=\mathbb{Z}$. Further we consider the exact homology sequence of the pair $\left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ and prove that its inclusion operator $i_*: H_k(\mathbb{C}\Omega_n) \rightarrow H_k(\mathbb{C}\Omega_{n+1})$ is zero. Taking into account that the relative homology groups $H_k \left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ are zero if $1\leq k \leq n$ and the inclusion operator $i_*=0$ we have derived from the exact homology sequence of the pair $\left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ that the homology groups $H_k \left ( \mathbb{C}\Omega_n \right ), 1\leq k

The space $\mathbb{C}\Omega_3$ is considered. The Betti numbers of the space $\mathbb{C}\Omega_3$ are calculated.

The spaces $\Omega_n(m)$ that generalize the spaces $\Omega_n$ are introduced. In order to investigate the homotopy invariants of the space $\Omega_n(m)$ the CW-structure of the space $\Omega_n(m)$ is built. Using exact homology sequence the homology groups of the space $\Omega_n(m)$ are calculated.

K. Borsuk in 1979, at the Topological Conference in Moscow, introduced concept of the capacity of a compactum and asked some questions concerning properties of the capacity ofcompacta. In this paper, we give partial positive answers to three of these questions in some cases. In fact, by describing spaces homotopy dominated by Moore and Eilenberg-MacLane spaces, the capacities of a Moore space $M(A,n)$ and an Eilenberg-MacLane space $K(G,n)$ could be obtained. Also, we compute the capacity of wedge sum of finitely many Moore spaces of different degrees and the capacity of product of finitely many Eilenberg-MacLane spaces of different homotopy types. In particular, we compute the capacity of wedge sum of finitely many spheres of the same or different dimensions.

In this paper we study an open problem: Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence? Since every ANR has the homotopy type of a polyhedron, this question is related in an obvious way to the longstanding Borsuk's problem (1967) Is it true that two ANR's homotopy dominating each other have the same homotopy type? The answer is positive for all polyhedra (or ANR's) with polycyclic-by-finite fundamental groups, manifolds, and clearly for 1-dimensional polyhedra. In some previous paper, we proved that this also holds for all 2-dimensional polyhedra with weakly Hopfian (hence Hopfian and co-Hopfian) fundamental groups (a group is weakly Hopfian if it is not isomorphic to a proper retract of itself). In this paper we extend this result to all (G , n) -complexes (where n ≥ 2). A (G , n) -complex, is an n -dimensional finite CW -complex with fundamental group G and all the homotopy groups π r in dimensions 1 < r < n trivial. We also discuss fundamental groups of potential counterexamples. As a corollary we obtain that for the same classes of spaces, the Borsuk's question: Is it true that the homotopy types of two quasi-homeomorphic ANR's are equal? , has positive answer (this question was also asked by S. Ferry in the recent edition of Scottish Book (1981)). [ABSTRACT FROM AUTHOR]

Let Pn be a simple n-polytope with a Z2-characteristic function λ. And h is a Morse function over Pn. Then the small cover Mn(λ) corresponding to the pair (Pn, λ) has a cell structure given by h. From this cell structure we can derive a cellular chain complex of Mn(λ) with integer coefficients. In this paper, firstly, we discuss the highest dimensional boundary morphism ∂n of this cellular chain complex and get that ∂n = 0 or 2 by a natural way. And then, from the well-known result that the submanifold corresponding to (F, λF) is naturally a small cover with dimension k, where F is any k-face of Pn and λF is the restriction of λ on F, we get that ∂k = 0 or ±2 for 0 ≤ k < n. Finally, by using the definition of inherited characteristic function which is the restriction of λ on the faces of Pn, we get a way to calculate the homology groups of Mn(λ). Applying our result to a 3-small cover we have that the homology groups of any 3-small cover is torsion-free or has only 2-torsion. [ABSTRACT FROM AUTHOR]

K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum. In this paper, we compute the capacity of wedge sum of finitely many spheres of various dimensions. [ABSTRACT FROM AUTHOR]

In 2012, Žiga Virk introduced the notion of an extensional equivalence , herein called an extensional map , and used it to generalize part of the extension theory factorization theorem of M. Levin, L. Rubin, and P. Schapiro. Here were are going to study this notion in the setting of inverse systems of compact Hausdorff spaces and approximate inverse systems of compact metric spaces. In both cases we will show that given a surjective map f to the limit, if each coordinate map p γ ∘ f is an extensional map, then so is f . [ABSTRACT FROM AUTHOR]

Let C be a class of spaces. An element Z ∈ C is called universal for C if each element of C embeds in Z . It is well-known that for each n ∈ N , there exists a universal element for the class of metrizable compacta X of (covering) dimension dim ⁡ X ≤ n . The situation in cohomological dimension over an abelian group G , denoted dim G , is almost the opposite. Our results will imply in contradistinction that for each nontrivial abelian group G and for n ≥ 2 , there exists no universal element for the class of metrizable compacta X with dim G ⁡ X ≤ n . [ABSTRACT FROM AUTHOR]

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graph Λ, this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions for k-graphs: projective limits and crossed products by finitely generated free abelian groups. [ABSTRACT FROM PUBLISHER]

Let Z, H be spaces. In previous work, we introduced the direct system X induced by the set of maps between the spaces Z and H. Now we will consider the case that X is induced by possibly a proper subset of the maps of Z to H. Our objective is to explore conditions under which X = dirlim X will be T, Hausdorff, regular, completely regular, pseudo-compact, normal, an absolute co-extensor for some space K, or will enjoy some combination of these properties. [ABSTRACT FROM AUTHOR]

The Lusternik-Schnirelmann category of a PL-manifold does not increase if we delete a point of it. This is false in the CW-complex category.

It has been conjectured that if $G= \mathop{({ \mathbb{Z} }_{p} )}\nolimits ^{r} $ acts freely on a finite $CW$-complex $X$ which is homotopy equivalent to a product of spheres ${S}^{{n}_{1} } \times {S}^{{n}_{2} } \times \cdots \times {S}^{{n}_{k} } $, then $r\leq k$. We address this question with the relaxation that $X$ is finite-dimensional, and show that, to answer the question, it suffices to consider the case where the dimensions of the spheres are greater than or equal to $2$. [ABSTRACT FROM AUTHOR]

Abstract: Let Z, H be spaces. We introduce the direct system X induced by the set of maps between the spaces Z and H and explore conditions under which will be pseudo-compact, but not necessarily compact. We also define the concept of a space Y being an absolute extensor relative to convergent sequences for a given space Z, which is useful in our situation and is used in formulating sufficient conditions for to be pseudo-compact. [Copyright &y& Elsevier]

Abstract: In the paper titled “Bockstein basis and resolution theorems in extension theory” (Tonić, 2010 ), we stated a theorem that we claimed to be a generalization of the Edwards–Walsh resolution theorem. The goal of this note is to show that the main theorem from Tonić (2010) is in fact equivalent to the Edwards–Walsh resolution theorem, and also that it can be proven without using Edwards–Walsh complexes. We conclude that the Edwards–Walsh resolution theorem can be proven without using Edwards–Walsh complexes. [Copyright &y& Elsevier]

Abstract: In some previous paper D. Kołodziejczyk (2001) we proved that there exists a finite polyhedron P with infinitely many left neighbors in the homotopy category, i.e. P homotopy dominates infinitely many finite polyhedra of different homotopy types and there isnʼt any homotopy type between P and . This answers a question of K. Borsuk (Borsuk, 1975 ). The dimension of that P is 3. Here we show that there exists a finite 2-dimensional polyhedron with the same property. Some remarks on constructing examples of 2-dimensional finite polyhedra dominating infinitely many different homotopy types are also included. [Copyright &y& Elsevier]

Abstract: Topologies on base point preserving function spaces are studied making use of the Brown–Booth–Tillotson -smash product and the topologies on function spaces defined by the classes of exponentiable spaces. Some conditions on the classes are obtained for exponential bijections and exponential homeomorphisms in the category of topological spaces with base point. Conditions for exponential homeomorphisms are obtained for CW-complexes whose cellular structure implies precise results. [ABSTRACT FROM AUTHOR]

Abstract: As the main results of this paper we prove that for every polyhedron P with abelian or torsion-free nilpotent fundamental group there are only finitely many different homotopy types of such that . The same holds for any finite with nilpotent fundamental group in place of . The problem, if there exists a polyhedron with infinitely many direct factors of different homotopy types (K. Borsuk, 1970) is still unsolved, even if we assume the second factor to be . [ABSTRACT FROM AUTHOR]

Abstract: We prove a generalization of the Edwards–Walsh Resolution Theorem: [Copyright &y& Elsevier]

Abstract: Let and be an n-dimensional CW-complex with the homotopy type of the n-sphere. We determine the automorphism group and then compute the number of distinct homotopy types of spherical space forms with respect to free and cellular G-actions on all CW-complexes , where 2d is a period of G. Next, the group of homotopy self-equivalences of spherical space forms associated with such G-actions α on are studied. Similar results for the rest of finite periodic groups have been obtained recently and they are described in the introduction. [Copyright &y& Elsevier]

We present new, unified proofs for the cell-like-, Z / p -, and Q -resolution theorems. Our arguments employ extensions that are much simpler than those used by our predecessors. The techniques allow us to solve problems involving cohomology groups by converting them into problems about homology groups. We provide a coordinated general topological method for constructing the maps needed to witness the resolution theorems simultaneously. [ABSTRACT FROM AUTHOR]

Expressions for the Lefschetz numbers of maps of fiber spaces in terms of those of maps of the fibers and the base are obtained. A similar problem is considered for Lefschetz coincidence numbers. [ABSTRACT FROM AUTHOR]

Abstract: We answer a question of K. Borsuk (1972) showing that there exists a connected polyhedron of dimension 4 with two different decompositions into a direct product of prime factors (in the homotopy or shape category). [Copyright &y& Elsevier]

If a monoid S is given by some finite complete presentation ℘, we construct inductively a chain of CW-complexes such that Δ n has dimension n, for every 2≤ m≤ n, the m-skeleton of Δ n is Δ m , and p m are critical ( m+1)-cells with 1≤ m≤ n−2. For every 2≤ m≤ n−1, the following is an exact sequence of (ℤ S,ℤ S)-bimodules where $(\mathcal{D},\mathbf{p}_{1},\ldots,\mathbf{p}_{m-2})=\mathcal{D}$ if m=2. We then use these sequences to obtain a free finitely generated bimodule partial resolution of ℤ S. Also we show that for groups properties FDT and FHT coincide. [ABSTRACT FROM AUTHOR]

Abstract: Suppose that K is a CW-complex. When we say that a space Y is an absolute co-extensor for K, we mean that K is an absolute extensor for Y, i.e., that for every closed subset A of Y and any map , there exists a map that extends f. Our main theorem will provide several statements that are equivalent to the condition that whenever K is a CW-complex and X is a space which is the topological sum of a countable collection of compact metrizable spaces each of which is an absolute co-extensor for K, then the Stone-Čech compactification of X is an absolute co-extensor for K. [Copyright &y& Elsevier]

We prove that if M is a CW-complex, then the homotopy type of the skeletal filtration of M does not depend on the cell decomposition of M up to wedge products with n-disks Dn, when the latter are given their natural CW-decomposition with unique cells of order 0, (n - 1) and n, a result resembling J.H.C. Whitehead's work on simple homotopy types. From the higher homotopy van Kampen Theorem (due to R. Brown and P.J. Higgins) follows an algebraic analogue for the fundamental crossed complex II(M) of the skeletal filtration of M, which thus depends only on the homotopy type of M (as a space) up to free product with crossed complexes of the type Dn ≐ II(Dn), n ϵ N. This expands an old result (due to J.H.C. Whitehead) asserting that the homotopy type of II(M) depends only on the homotopy type of M. We use these results to define a homotopy invariant IA of CW-complexes for each finite crossed complex A. We interpret it in terms of the weak homotopy type of the function space TOP (((M, *), (∣A∣,*), where ∣A∣ is the classifying space of the crossed complex [ABSTRACT FROM AUTHOR]

Let G be a finite group given in one of the forms listed in the title with period 2d and X(n) an n-dimensional CW-complex with the homotopy type of an n-sphere. We study the automorphism group Aut (G) to compute the number of distinct homotopy types of orbit spaces X(2dn — 1)/μ with respect to free and cellular G-actions μ on all CW — complexes X(2dn—1). At the end, the groups ϵ(X(2dn—1)/μ) of self homotopy equivalences of orbit spaces X(2dn — 1)/μ associated with free and cellular G-actions μ on X(2dn — 1) are determined. [ABSTRACT FROM AUTHOR]

Abstract: For any graph without loops, let denote the regular CW-complex obtained from G by attaching to each circuit C of G a disc. We show that if G is the suspension of a flat graph, then has an embedding into 4-space. Furthermore, we show that for any graph G in the collection of graphs that can be obtained from and by a series of - and -transformations, cannot be embedded into 4-space. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a novel shape and physical attribute handling system Volume-CAD designated for practical and robust use in the manufacturing process of various industries. The internal data structure ‘Kitta Cube’ is formalized by CW-complexes. Complex is a structure consisting of different dimensional items with consistency (without gap or overlaps), therefore allowing each volume comprising of a material separated by boundaries to be represented completely. Volume-CAD (VCAD 1 [1] Registered trade mark. ) could be called ‘solid and fluid model’, as it goes beyond the solid model. As applications of VCAD, this paper discusses surface simplification of Kitta Cubes, cell labeling (three-dimensional coloring), and multiple-materials situation. These models can be used not only for manufacturing but also throughout the product life cycle from shipment tracking until the end of the product life cycle. [Copyright &y& Elsevier]

Abstract: In the previous papers, in connection with a question of K. Borsuk, we proved that there exist polyhedra with polycyclic fundamental groups homotopy dominating infinitely many different homotopy types. Here we consider a few problems of K. Borsuk concerning infinite chains of polyhedra or FANR''s ordered by the relation of domination (in homotopy or shape category) and obtain that for polyhedra with polycyclic-by-finite fundamental groups, there are no pathology similar to the above. [Copyright &y& Elsevier]

In 1968 K. Borsuk asked: Is it true that every finite polyhedron dominates only finitely many different shapes? In this question the notions of shape and shape domination can be replaced by the notions of homotopy type and homotopy domination.We obtained earlier a negative answer to the Borsuk question and next results that the examples of such polyhedra are not rare. In particular, there exist polyhedra with nilpotent fundamental groups dominating infinitely many different homotopy types. On the other hand, we proved that every polyhedron with finite fundamental group dominates only finitely many different homotopy types. Here we obtain next positive results that the same is true for some classes of polyhedra with Abelian fundamental groups and for nilpotent polyhedra. Therefore we also get that every finitely generated, nilpotent torsion-free group has only finitely many r-images up to isomorphism. [Copyright &y& Elsevier]

Let G be a finite group one of the forms given in the title with period 2d and X(n) an n-dimensional CW-complex with the homotopy type of an n-sphere.We study the automorphism group Aut(G) to compute the number of distinct homotopy types of spherical space forms with respect to G-actions on all CW-complexes X(2dn-1). At the end, the groups E(X(2dn-1)/γ) of self homotopy equivalences of space forms X(2dn-1)/γ associated with free and cellular G-actions γ on X(2dn-1) are determined. [Copyright &y& Elsevier]

In 1994, R. Cauty proved Geoghegan''s conjecture on ANRs that a metrizable space X is an ANR if and only if every open set in X has the homotopy type of a CW-complex. In this note, by using the mapping cylinder technique, we provide an alternative short proof of this characterization of ANRs. [Copyright &y& Elsevier]

Extension dimension is characterized in terms of ω-maps. We apply this result to prove that extension dimension is preserved by refinable maps between metrizable spaces. It is also shown that refinable maps preserve some infinite-dimensional properties. [ABSTRACT FROM AUTHOR]