The relation between homotopy limits and Bauer's shape singular complex
Hanns Thiemann (1989)
Cahiers de Topologie et Géométrie Différentielle Catégoriques
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Hanns Thiemann (1989)
Cahiers de Topologie et Géométrie Différentielle Catégoriques
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R. Goad (1983)
Fundamenta Mathematicae
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A. Suszycki (1983)
Fundamenta Mathematicae
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Stanisław Godlewski (1972)
Fundamenta Mathematicae
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Zvonko Cerin (1994)
Collectanea Mathematica
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Sibe Mardešić, Jack Segal (1971)
Fundamenta Mathematicae
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J. Quigley (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.
D. Coram, P. Duvall (1980)
Fundamenta Mathematicae
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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.
Luciano Stramaccia (2002)
Cahiers de Topologie et Géométrie Différentielle Catégoriques
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