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Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

Friedhelm Waldhausen
Bjørn Jahren
John Rognes
Copyright Date: 2013
Pages: 184
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  • Book Info
    Spaces of PL Manifolds and Categories of Simple Maps (AM-186)
    Book 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.

    eISBN: 978-1-4008-4652-8
    Subjects: Mathematics

Table of Contents

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  1. Front Matter
    (pp. [i]-[iv])
  2. Table of Contents
    (pp. [v]-[viii])
  3. Introduction
    (pp. 1-6)

    We present a proof of the stable parametrizedh-cobordism theorem, which we choose to state as follows:

    Theorem 0.1.There is a natural homotopy equivalence$\mathcal{H}^{CAT}(M)\simeq \Omega\,\mathrm{Wh}^{CAT}(M)$for each compact CAT manifold M, with CAT = TOP, PL or DIFF.

    Here$\mathcal{H}^{CAT}(M)$(M) denotes a stable CATh-cobordism space defined in terms of manifolds, whereas WhCAT(M) denotes a CAT Whitehead space defined in terms of algebraicK-theory. We specify functorial models for these spaces in Definitions 1.1.1, 1.1.3, 1.3.2 and 1.3.4. See also Remark 1.3.1 for comments about our notation.

    This is a stable range extension to parametrized families of the...

  4. Chapter One The stable parametrized h-cobordism theorem
    (pp. 7-28)

    We write DIFF for the category ofCsmooth manifolds, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. We generically write CAT for any one of these geometric categories. LetI= [0, 1] andJbe two fixed closed intervals in$\mathbb{R}$. We will form collars usingIand stabilize manifolds and polyhedra usingJ.

    In this section, as well as in Chapter 4, we let$\Delta^{q}=\{(t_{0},\ldots,t_{q})|\sum_{i=0}^{q}t_{i}=1,t_{i}\geq 0\}$be the standard affineq-simplex.

    By a CAT bundle$\pi:E\rightarrow\Delta^{q}$we mean a CAT locally trivial family, i.e., a map such that there exists...

  5. Chapter Two On simple maps
    (pp. 29-98)

    In this section we define simple maps of finite simplicial sets, and establish some of their formal properties.

    Let Δ be the skeleton category of finite non-empty ordinals, with objects the linearly ordered sets [n] = {0 < 1 < ⋯ <n} forn≥ 0, and morphismsα: [n] → [m] the order-preserving functions. A simplicial setXis a contravariant functor from Δ to sets. The simplicial set Δqis the functor represented by the object [q]. By a simplex inXwe mean a pair ([n],x), wheren≥ 0 andxXn, but we shall usually...

  6. Chapter Three The non-manifold part
    (pp. 99-138)

    In this chapter, let Δqdenote the simplicialq-simplex, the simplicial set represented by [q] in Δ, and let |Δq| denote its geometric realization, the standard affineq-simplex.

    Definition 3.1.1. Let$\mathcal{C}$be the category of finite simplicial setsXand simplicial mapsf:XY. Let$\mathcal{D}$be the full subcategory of$\mathcal{C}$of finite non-singular simplicial sets, and write$i: \mathcal{D}\rightarrow \mathcal{C}$for the inclusion functor. Let$\mathcal{E}$be the category of compact polyhedraKand PL mapsf:KL, and write$r: \mathcal{D}\rightarrow \mathcal{E}$for the polyhedral realization functor. We also writeiandr...

  7. Chapter Four The manifold part
    (pp. 139-174)

    In this chapter, let Δqdenote the standard affineq-simplex. All polyhedra will be compact, and all manifolds considered will be compact PL manifolds, usually without further mention. Recall the fixed intervalsIandJfrom Section 1.1.

    The aim of this section is to reduce the proof of the manifold part of the stable parametrizedh-cobordism theorem, Theorem 1.1.8, to a result about spaces of stably framed manifolds, Theorem 4.1.14, which will be proved in the following two sections.

    Definition 4.1.1. LetPbe a compact polyhedron. By a family of manifolds parametrized byPwe shall mean a...

  8. Bibliography
    (pp. 175-178)
  9. Symbols
    (pp. 179-180)
  10. Index
    (pp. 181-184)