Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 521finite–dimensions.The Atiyah–Singer index is then defined byInd(T ) = dim[Ker(T )] − dim[Coker(T )],and it has the key property that it is continuous, and therefore constant oneach connected component of F(H). Moreover, Ind : F → Z identifiesthe components of F.This has a generalization to any compact space X. To any continuousmap X → F (i.e., a continuous family of Fredholm operators parametrizedby X) one can assign an index in K(X). Moreover one gets in this wayan isomorphism Ind : [X, F] → K(X), where [X, F] denotes the set ofhomotopy classes of maps of X into F. A notable example of a Fredholmoperator is an elliptic differential operator on a compact manifold (theseare turned into bounded operators by using appropriate Sobolev norms).Now, in the quantum–physical situation, one meets infinite–order elementsα ∈ H 3 (X, Z) and the question arises of whether one can still definea ‘twisted’ group K α (X). It turns out that it is possible to do this and oneapproach is being developed by [Atiyah and Segal (1971)].Since an α of order n arises from an obstruction problem involvingnD vector bundles, it is plausible that, for α of infinite order, we need toconsider bundles of Hilbert spaces H. But here we have to be careful notto confuse the ‘small’ unitary groupU(∞) = limN→∞ U(N)with the ‘large’ group U(H) of all unitary operators in Hilbert space. Thesmall unitary group has interesting homotopy groups given by Bott’s periodicityTheorem, but U(H) is contractible, by Kuiper’s Theorem. Thismeans that all Hilbert space bundles (with U(H)) as structure group) aretrivial. This implies the following homotopy equivalences:P U(H) = U(H)/U(1) ∼ CP ∞ = K(Z, 2), BP U(H) ∼ K(Z, 3),where B denotes here the classifying space and on the right we have theEilenberg–MacLane spaces. It follows that P (H)−bundles over X are classifiedcompletely by H 3 (X, Z). Thus, for each α ∈ H 3 (X, Z), there is anessentially unique bundle P α over X with fibre P (H).As in finite dimensions. B(H) depends only on P (H), we can define abundle B α of algebras over X. We now let F α ⊂ B α be the correspondingbundle of Fredholm operators. Finally we defineK α (X) = Homotopy classes of sections of F α .