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This book presents a proof of the stable parametrized h-cobordism theorem, which deals with the existence of a natural homotopy equivalence for each compact CAT manifold. In this theorem, a stable CAT h-cobordism space is defined in terms of manifolds, whereas a CAT Whitehead space is defined in terms of algebraic K-theory. This is a stable range extension to parametrized families of the classical hand s-cobordism theorems first stated by A. E. Hatcher, but his proofs were incomplete. This book provides a full proof of this key result, which provides the link between the geometric topology of high-dimensional manifolds and their automorphisms, as well as the algebraic K-theory of spaces and structured ring spectra.

This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. CAT is generically written for any one of these geometric categories. Relevant terms such as stabilization map, simple map, pullback map, PL Serre fibrations, weak homotopy equivalence, PL Whitehead space, and cofibration are also defined. The chapter proceeds by describing the non-manifold part, the algebraic K-theory of spaces, and the relevance of simple maps to the study of PL homeomorphisms of manifolds.

This chapter discusses some of the techniques that enter into L theory. The nature of L groups for groups π‎ is very different if π‎ is finite or if π‎ is torsion-free. The latter case seems to be governed by rigidity phenomena and is conjecturally very closely related to group homology. Finite groups, however, are studied algebraically. The chapter then explains some of the mathematics that surround the calculation of L groups, looking at connections to Witt theory, algebraic K theory, and representation theory. The connection to representation theory motivates Dress induction. The chapter considers Shaneson's thesis which shows that the Farrell fibering theorem is equivalent to a calculation of L groups.

In this chapter we present some of the physical processes that are used in numerical weather prediction modeling. Grid-point models, based on finite differences, and spectral models both generally treat the physical processes in the same manner. The vertical columns above the horizontal grid points (the transform grid for the spectral models) are the ones along which estimates of the effects of the physical processes are made. In this chapter we present a treatment of the planetary boundary layer, including a discussion on the surface similarity theory. Also covered is the cumulus parameterization problem in terms of the Kuo scheme and the Arakawa- Schubert sheme. Large-scale condensation and radiative transfer in clear and cloudy skies are the final topics reviewed. There are at least three types of fluxes that one deals with, namely momentum, sensible heat, and moisture. Furthermore, one needs to examine separately the land and ocean regions. In this section we present the socalled bulk aerodynamic methods as well as the similarity analysis approach for the estimation of the surface fluxes. The radiation code in a numerical weather prediction model is usually coupled to the calculation of the surface energy balance. This will be covered later in Section 8.5.6. This surface energy balance is usually carried out over land areas, where one balances the net radiation against the surface fluxes of heat and moisture for the determination of soil temperature. Over oceans, the sea-surface temperatures are prescribed where the surface energy balance is implicit. Thus it is quite apparent that what one does in the parameterization of the planetary boundary layer has to be integrated with the radiative parameterization in a consistent manner.