process

Twisted K-theory – old and new 129Lemma 3.4. Any element of GrK 1 .ƒ/ may be written as the homotopy class of apair .E; "/ where E is a free graded B-module and " is a grading of degree 1 of theassociated ƒ-module (see 1.4 for the definition of a grading).By the well-known dictionary between modules and bundle theory, we may view " asa grading of a suitable bundle of free End.H /=K-modules. By spectral theory, we mayalso assume that " is self-adjoint. Finally, following [29], we define a quasi-grading 16of E as a family of Fredholm endomorphisms D such that1. D D D,2. D is of degree 1.The following theorem is the analogue in the graded case of Theorem 2.8 (cf [29],p. 78/79).Theorem 3.5. The (graded) twisted K-group K A .X/ is the Grothendieck group associatedto the semi-group of homotopy classes of pairs .E; D/ where E isafreeZ=2-graded B-module and D is a family of Fredholm endomorphisms of E which areself-adjoint and of degree 1.Remark 3.6. Let us assume that A is oriented modelled on M 2 .K/. The descriptionabove gives a Fredholm description of GrK 1 .A/ D K 1 .A/: we just take the homotopyclasses of sections of the associated bundle of self-adjoint Fredholm operatorsFredh .B/ whose essential spectrum is divided into two non empty parts 17 in R C andR .This Fredholm description of K A .X/ enables us to define a cup-productK A .X/ K A0 .X/ KA y˝A 0 .X/where A y˝ A 0 denotes the graded tensor product of A and A 0 . This cup-product isgiven by the same formula as in [19], p. 19, and generalizes it:.E; D/ Y .E 0 ;D 0 / D .E y˝ E 0 ;D y˝ 1 C 1 y˝ D 0 /(see the appendix about the origin of this formula in usual topological K-theory).To conclude this section, let us consider a locally compact space X and a bundle ofgraded algebras A on X. For technical reasons, we assume the existence of a compactspace Z containing X as an open subset, such that A extends to a bundle (also calledA) onZ. There is an obvious definition of K A .X/ as a relative term in the followingexact sequence (where T D Z X and A 0 D A y˝ C 0;1 ):K A0 .Z/ KA 0 .T / K A .X/ K A .Z/ K A .T /.16 “quasi-graduation” in French.17 See the appendix about the role of self-adjoint Fredholm operators in K-theory.