Shape and strong shape equivalences
Luciano Stramaccia (2002)
Cahiers de Topologie et Géométrie Différentielle Catégoriques
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Luciano Stramaccia (2002)
Cahiers de Topologie et Géométrie Différentielle Catégoriques
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Peter Mrozik (1991)
Compositio Mathematica
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Zvonko Cerin (1994)
Collectanea Mathematica
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Marek Golasiński, Luciano Stramaccia (2008)
Cahiers de Topologie et Géométrie Différentielle Catégoriques
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Danuta Kołodziejczyk (2003)
Fundamenta Mathematicae
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In 1968 K. Borsuk asked: Does every polyhedron dominate only finitely many different shapes? In this question the notion of shape can be replaced by the notion of homotopy type. We showed earlier that the answer to the Borsuk question is no. However, in a previous paper we proved that every simply connected polyhedron dominates only finitely many different homotopy types (equivalently, shapes). Here we prove that the same is true for polyhedra with finite fundamental group.
Sibe Mardešić (1976)
Fundamenta Mathematicae
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Danuta Kołodziejczyk (2003)
Fundamenta Mathematicae
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We show the existence of a finite polyhedron P dominating infinitely many different homotopy types of finite polyhedra and such that there is a bound on the lengths of all strictly descending sequences of homotopy types dominated by P. This answers a question of K. Borsuk (1979) dealing with shape-theoretic notions of "capacity" and "depth" of compact metric spaces. Moreover, π₁(P) may be any given non-abelian poly-ℤ-group and dim P may be any given integer n ≥ 3.
Luc Demers (1975)
Fundamenta Mathematicae
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Y. Kodama, J. Ono (1979)
Fundamenta Mathematicae
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M.A. Morón, F.R. Ruiz del Portal (1993)
Manuscripta mathematica
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