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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]

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]

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: 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 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]

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]