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

Abstract

Since its introduction by the author 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 the author'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.

Quillen's work in algebraicK-theory

“…Surveys of that material include [50] and the more recent [82]. The statement and the careful and detailed proof of the stable parametrized h-cobordism theorem in [79] depend on the algebraic K-theory of topological spaces to relate the geometric topology of manifolds to algebraic K-theory in a succinct and powerful form. Waldhausen's basic theorems also laid the foundation for Thomason's extension of Quillen's K-theory that incorporates chain complexes naturally, leading to his localization theorem [74] for schemes.…”

On transfer maps in the algebraic K-theory of spaces

“…where Q X + : = ∞ ∞ X + and η X denotes the unit transformation. If X is a compact smooth manifold, the space Wh Diff (X ) is homotopy equivalent to the stable parametrized h-cobordism space of X [22]. The unit transformation admits a natural retraction up to homotopy, given by the Waldhausen trace map, Tr X : A(X ) → Q X + , and therefore we obtain from (1) a natural homotopy equivalence of infinite loop spaces: In addition to the covariant functoriality, A-theory also admits transfer maps for fibrations p : E → B with homotopy finite fibers.…”

Corrigendum to: “A model structure à la Thomason on 2-Cat” [J. Pure Appl. Algebra 208 (2007) 205–236]

“…Let X be a non-singular simplicial set, i.e., a simplicial set such that the simplicial map ∆ n → X representing any non-degenerate simplex is a cofibration of simplicial sets [8,Definition 1.2.2]. It follows from [8, Lemmas 2.2.9 and 2.2.11] that if X is non-singular, then the second simplicial subdivision Sd 2 (X) of X can be calculated as a Barratt-nerve…”