CHAPTER 2

K-Theory with Reality

This chapter provides a number of the basic tools we need in the rest of the

book. Much of this summarizes and extends classical material in a convenient form.

Thus Sections 2.1 and 2.2 recall basic representation theory and periodic K-theory

with an involution in the sense of Atiyah and Karoubi. In Section 2.3 we give an

extended summary of the facts we need from Clifford algebras.

This equips us to describe in Section 2.4 the adaption of the construction of

a fully equivariant version of the connective theory [46] to the case with real-

ity. This provides the basis for the fully equivariant (uncompleted) version of our

calculations. Sections 2.5 and 2.6 prove and exploit the local cohomology and com-

pletion theorems which provide the basic connections to non-equivariant theories.

In Section 2.7 we show how well suited this is to the study of the Gromov-Lawson-

Rosenberg conjecture on the existence of positive scalar curvature metrics on spin

manifolds.

Finally, we consider the first few negative coefficient groups of the theory. By

using the classical results of Bott (recalled in Section 2.8) and Atiyah-Hirzebruch

we are able to show in Section 2.9 that kon G in terms of representation theory for

n≥-7.

2.1. Representation theory

We summarize here the standard representation theory that we need. See

[4, 22] for further details.

The basic structures are linear representations.

Definition 2.1.1. When K = R, C or H a representation of G over K is a

K-vector space V with a K-linear action of G on V .

For the purpose of comparison, and for the study of multiplicative structure, it

is worth giving an alternative approach.

Definition 2.1.2. (i) A real representation of G is a representation V of G

over C with a conjugate linear map J : V -→V with J2 = 1.

(ii) A quaternionic representation of G is a representation V of G over C with a

conjugate linear map J : V -→V with J2 = -1.

(iii) A complex representation of G is the same as a representation over C.

Warning 2.1.3. To avoid confusion later, we emphasize that a real represen-

tation is a different thing to a representation over R. Similarly a quaternionic

representation is a different thing to a representation over H.

Lemma 2.1.4. (i) There is an equivalence of categories between representations

over R and real representations. A representation U over R gives rise to the real

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http://dx.doi.org/10.1090/surv/169/02