Geometric realization of homotopy systems.
Item
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Title
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Geometric realization of homotopy systems.
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Identifier
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AAI9530872
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identifier
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9530872
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Creator
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Goldstone, Richard Jay.
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Contributor
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Adviser: Alex Heller
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Date
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1995
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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In this work, we study the geometric realization problem for homotopy systems. Given a crossed complex C of free type, we construct a CW complex KC, unique up to homotopy, that in addition to its CW filtration has a filtration {dollar}\{lcub}J\sp{lcub}n{rcub}KC\{rcub}{dollar} that is, in general, different than the CW filtration. The filtered space {dollar}KC\sb{lcub}\rm J{rcub}{dollar} has a homotopy system{dollar}{dollar}\eqalign{lcub}\rho(KC\sb{lcub}J{rcub}) = \cdots \to\pi\sb{lcub}k{rcub}(J\sp{lcub}k{rcub}KC,& J\sp{lcub}k-1{rcub}KC)\to\pi\sb{lcub}k-1{rcub}\cr&\quad\quad\sk{lcub}45{rcub}(J\sp{lcub}k-1{rcub}KC, J\sp{lcub}k-2{rcub}KC)\to\cr&\cdots\to\pi\sb2(J\sp2\ KC, J\sp1\ KC)\to\pi\sb1(KC\sp1){rcub}{dollar}{dollar} that realizes C. There is a map {dollar}\alpha:\rho(KC\sb{lcub}\rm CW{rcub})\to C{dollar} whose chain homotopy class contains a chain epimorphism {dollar}\bar\alpha{dollar}. Our main result is that the existence of a chain splitting of this epimorphism is a necessary and sufficient condition for geometric realization; in other words, C can be geometrically realized if and only if it is a retract of the crossed complex {dollar}\rho(KC\sb{lcub}\rm CW{rcub}{dollar}). The space KC has the homotopy type of the classifying space of a crossed complex BC, obtained by completely different methods by Brown and Higgins.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
