Ivancevic_Applied-Diff-Geom

518 Applied Differential Geometry: A Modern IntroductionIn addition to the K(X)−module structure of K α (X) there are multiplicationsK α (X) ⊗ K β (X) → K α+β (X).Today, there are many variants and generalizations of K−theory, somethingwhich is not surprising given the universality of linear algebra andmatrices [Atiyah and Anderson (1967); Atiyah (2000)]. In each case thereare specific features and techniques relevant to the particular area.First, as already mentioned, is the real K−theory based on real vectorbundles and the Bott periodicity theorems for the orthogonal groups: herethe period is 8 rather than 2.Next there is equivariant theory K G (X), where G is a a compact Liegroup acting on the space X. If X is a point, we just get the representationor character ring R(G) of the group G. In general K G (X) is a moduleover R(G) and this can be exploited in terms of the fixed–point sets in Xof elements of G.If we pass from the space X to the ring C(X) of continuous complex–valued functions on X then K(X) can be defined purely algebraically interms of finitely–general projective modules over X. This then lends itselfto a major generalization if we replace C(X), which is a commutativeC ∗ −algebra, by a non–commutative C ∗ −algebra. This has become a richtheory linked to many basic ideas in functional analysis, in particular tothe von Neumann dimension theory.4.5.5 Atiyah–Singer Index TheoremWe shall recall here very briefly some essential results of Atiyah–SingerIndex Theory. The reader who is not familiar with the topological and analyticproperties of the index of elliptic operators is urged to gain some familiaritywith the Atiyah–Singer Index Theorem [Atiyah and Singer (1963);Atiyah and Singer (1968)] 8 (for technical details, see also [Boos and Bleecker8 In the geometry of manifolds and differential operators, the Atiyah—Singer IndexTheorem is an important unifying result that connects topology and analysis. It dealswith elliptic differential operators (such as the Laplacian) on compact manifolds. It findsnumerous applications, including many in theoretical physics. When Michael Atiyah andIsadore Singer were awarded the Abel Prize by the Norwegian Academy of Science andLetters in 2004, the prize announcement explained the Atiyah—Singer Index Theoremin these words:“Scientists describe the world by measuring quantities and forces that vary overtime and space. The rules of nature are often expressed by formulas, called