The insufficiency of barycentric subdivision.

Finney, Ross Lee

doi:10.1307/mmj/1028999363

Cited by 6 publications

(13 citation statements)

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“…Prove that in this case every isomorphism g on the right-hand side is of the form g =/' for some isomorphism on the left-hand side. (Finney, 1965) 9. Show for simplicial complexes K,L : K^Lo(K') l^{ L) 1 .…”

Section: P(s) = {Px(s) • Xea}mentioning

confidence: 99%

Cellular Structures in Topology

1990

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This book describes the construction and the properties of CW-complexes. These spaces are important because firstly they are the correct framework for homotopy theory, and secondly most spaces that arise in pure mathematics are of this type. The authors discuss the foundations and also developments, for example, the theory of finite CW-complexes, CW-complexes in relation to the theory of fibrations, and Milnor's work on spaces of the type of CW-complexes. They establish very clearly the relationship between CW-complexes and the theory of simplicial complexes, which is developed in great detail. Exercises are provided throughout the book; some are straightforward, others extend the text in a non-trivial way. For the latter; further reference is given for their solution. Each chapter ends with a section sketching the historical development. An appendix gives basic results from topology, homology and homotopy theory. These features will aid graduate students, who can use the work as a course text. As a contemporary reference work it will be essential reading for the more specialized workers in algebraic topology and homotopy theory.

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Section: P(s) = {Px(s) • Xea}mentioning

confidence: 99%

Cellular Structures in Topology

1990

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“…L. Finney [1] proved that two locally-finite simplicial complexes are isomorphic if their first derived complexes are isomorphic. J. Segal [3] making use of this showed that two locally-finite simplicial complexes are isomorphic if the 1-sections of their first derived complexes are isomorphic.…”

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confidence: 99%

“…For nontriviality we assume each of M and N has at least 3 vertices. Let T M denote the set of vertices of mM whose order in | (mM) 1 \ is not 2 and let T N be the corresponding set in nN; then m is in T M and n is in T N .…”

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confidence: 99%

“…Now let h be a homeomorphism of \(mMy\ onto | (^ΛΓ) 1 1. We shall modify h where necessary to get a new homeomorphism h which maps the vertices of (mM) 1 onto those of (nN) 1 .…”

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confidence: 99%

“…If h(v) is a vertex of (nN) 1 we let h x = h. Suppose then that h(v) is not a vertex. Let C be the closure in | (mM) 1 \ of the component Q of | (mM) 1 \ -T M containing v\ then C is either an arc with endpoints in T M or a simple closed curve which one easily shows must be of the form Q U {m}.…”

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confidence: 99%

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