Automorphisms of manifolds

Abstract

This survey is about homotopy types of spaces of automorphisms of topological and smooth manifolds. Most of the results available are relative, i.e., they compare different types of automorphisms.In chapter 1, which motivates the later chapters, we introduce our favorite types of manifold automorphisms and make a comparison by (mostly elementary) geometric methods. Chapters 2, 3, and 4 describe algebraic models (involving L-theory and/or algebraic K-theory) for certain spaces of "structures" associated with a manifold M , that is, spaces of other manifolds sharing some geometric features with M . The algebraic models rely heavily onWall's work in surgery theory, e.g. [Wa1] , • Waldhausen's work in h-cobordism theory alias concordance theory, which includes a parametrized version of Wall's theory of the finiteness obstruction, [Wa2] .The structure spaces are of interest for the following reason. Suppose that two different notions of automorphism of M are being compared. Let X 1 (M ) and X 2 (M ) be the corresponding automorphism spaces; suppose that X 1 (M ) ⊂ X 2 (M ). As a rule, the space of cosets X 2 (M )/X 1 (M ) is a union of connected components of a suitable structure space.Chapter 5 contains the beginnings of a more radical approach in which one tries to calculate the classifying space BX 1 (M ) in terms of BX 2 (M ), rather than trying to calculate X 2 (M )/X 1 (M ). Chapter 6 contains some examples and calculations.

Obstructions to stably fibering manifolds

“…By our assumptions the manifold @ x F b is smoothable, so Igusa's stability result [17] (see Weiss and Williams [34,Theorem 1.3.4] for the topological range) says that…”

Homotopy-theoretically enriched categories of noncommutative motives

“…The homomorphism O(n) → Diff(D 0n ) is a homotopy equivalence, while the constructions of ( [72], § 3.2) define a system…”

Quillen's work in algebraicK-theory

“…Then the homotopy group π i Diff(S n ) has rank 0 if i does not have the form 4k − 1, it has rank 1 if n is even and i has the form 4k − 1, and it has rank 2 if n is odd and i has the form 4k − 1. 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.…”