CHAPTER 1

Introduction

The aim of this book is to study the connective real

K-cohomologyko*(BG)

as

a ring and the associated homology ko*(BG) as a module over the ring, for a range

of compact Lie groups G, with both general results and a wide range of specific

calculations. We believe that these calculations are only the beginning of what can

be done with these methods, and we hope others will find them useful in many

more cases.

1.1. Motivation

There are general reasons for interest that would not justify a project of this

size. On the one hand, calculations of connective K-theory of a space are illumi-

nating for its homotopy theory; on the other, classifying spaces BG provide both

embodiments of group theory and especially accessible infinite spaces.

A more compelling argument comes from three processes for deducing subtle

information from cruder information: uncompletion, descent and unlocalization.

All three occur rather widely, and most fashionably in the study of elliptic coho-

mology, but they still have considerable substance even in the more accessible case

of K-theory. To explain further, we imagine we live in a perfect world but with

imperfect understanding. Thus, although there is an equivariant form of connective

real K-theory, we begin with a knowledge of non-equivariant, periodic, complex K-

theory alone. Since the Atiyah-Segal theorem states that KU *(BG) is a completion

of KUG, * the process of going from a non-equivariant theory to the equivariant the-

ory can be viewed as uncompletion. Since KO is the fixed point spectrum of KU,

the process of recovering real from complex K-theory is a form of descent. Finally,

the periodic theory KO is obtained from the connective theory ko by inverting the

Bott element, so the recovery of ko from KO is an unlocalization. In the forms

just described, these processes are well understood theoretically, but if we ask for

equivariant forms of descent and unlocalization there is more to be said, and simi-

larly if there is a space involved as well. Finally, we might actually want to use this

theoretical understanding to give calculations. The present book could be viewed

as a case study in these matters.

Finally, it is worth explaining why we actually began this project. When G is

a discrete group, kon(BG) is the obstruction group for the existence of a positive

scalar curvature metric on a spin manifold of dimension n ≥ 5 and fundamental

group G. We will give a brief description of work on the Gromov-Lawson-Rosenberg

(GLR) conjecture, referring to [88] for a more detailed guide. First, for a spin

manifold M of dimension n≥ 5, the existence of a positive scalar curvature metric

depends only on the bordism class [M], and there are maps

Ωn

spin(BG)

→kon(BG)→KOn(BG)→KOn(CredG),*ApD

1

http://dx.doi.org/10.1090/surv/169/01