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Spaces of PL Manifolds and Categories of Simple Maps

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Title: Spaces of PL Manifolds and Categories of Simple Maps
Description: Since its introduction by Friedhelm Waldhausen in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a'desingularization,'improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.
Authors: Friedhelm Waldhausen, Bjørn Jahren, John Rognes
Resource Type: eBook.
Subjects: Piecewise linear topology, Mappings (Mathematics)
Categories: MATHEMATICS / Topology, MATHEMATICS / Transformations
Database: eBook Collection (EBSCOhost)
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  – Type: ebook-pdf
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  Availability: 0
Header DbId: nlebk
DbLabel: eBook Collection (EBSCOhost)
An: 529479
RelevancyScore: 1051
AccessLevel: 6
PubType: eBook
PubTypeId: ebook
PreciseRelevancyScore: 1050.81640625
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Spaces of PL Manifolds and Categories of Simple Maps
– Name: Abstract
  Label: Description
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  Data: Since its introduction by Friedhelm Waldhausen in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a'desingularization,'improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.
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  Data: <searchLink fieldCode="AR" term="%22Friedhelm+Waldhausen%22">Friedhelm Waldhausen</searchLink><br /><searchLink fieldCode="AR" term="%22Bjørn+Jahren%22">Bjørn Jahren</searchLink><br /><searchLink fieldCode="AR" term="%22John+Rognes%22">John Rognes</searchLink>
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  Data: <searchLink fieldCode="DE" term="%22Piecewise+linear+topology%22">Piecewise linear topology</searchLink><br /><searchLink fieldCode="DE" term="%22Mappings+%28Mathematics%29%22">Mappings (Mathematics)</searchLink>
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RecordInfo BibRecord:
  BibEntity:
    Classifications:
      – Code: 514.22
        Scheme: ddc
        Type: prePub
    Languages:
      – Code: eng
        Text: English
    Subjects:
      – SubjectFull: Piecewise linear topology
        Type: general
      – SubjectFull: Mappings (Mathematics)
        Type: general
    Titles:
      – TitleFull: Spaces of PL Manifolds and Categories of Simple Maps
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Friedhelm Waldhausen
      – PersonEntity:
          Name:
            NameFull: Bjørn Jahren
      – PersonEntity:
          Name:
            NameFull: John Rognes
      – PersonEntity:
          Name:
            NameFull: Friedhelm Waldhausen
      – PersonEntity:
          Name:
            NameFull: Bjørn Jahren
      – PersonEntity:
          Name:
            NameFull: John Rognes
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      – BibEntity:
          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2013
            – D: 04
              M: 02
              Type: profile
              Y: 2014
          Identifiers:
            – Type: isbn-print
              Value: 9780691157764
            – Type: isbn-print
              Value: 9780691157757
            – Type: isbn-electronic
              Value: 9781400846528
            – Type: isbn-electronic
              Value: 9781400847587
          Numbering:
            – Type: volume
              Value: 00186
          Titles:
            – TitleFull: Spaces of PL Manifolds and Categories of Simple Maps
              Type: main
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