Ivancevic_Applied-Diff-Geom

512 Applied Differential Geometry: A Modern IntroductionIf we describe the same vector bundle on a manifold in two differentways, the Chern classes will be the same, i.e., if the Chern classes of apair of vector bundles do not agree, then the vector bundles are different(the converse is not true, though). In topology, differential geometry,and algebraic geometry, it is often important to count how many linearlyindependent sections a vector bundle has. The Chern classes offer someinformation about this through, for instance, the Riemann–Roch Theoremand the Atiyah-Singer Index Theorem. Chern classes are also feasible tocalculate in practice. In differential geometry (and some types of algebraicgeometry), the Chern classes can be expressed as polynomials in the coefficientsof the curvature form.In particular, given a complex hermitian vector bundle V of complexrank n over a smooth manifold M, a representative of each Chern class(also called a Chern form) c k (V ) of V are given as the coefficients of thecharacteristic polynomial( ) itΩdet2π + I = c k (V )t k ,reason why a ‘dual’ theory to homology was sought. The characteristic class approachto curvature invariants was a particular reason to make a theory, to prove a generalGauss–Bonnet Theorem.When the theory was put on an organized basis around 1950 (with the definitionsreduced to homotopy theory) it became clear that the most fundamental characteristicclasses known at that time (the Stiefel–Whitney class, the Chern class, and the Pontryaginclass) were reflections of the classical linear groups and their maximal torusstructure. What is more, the Chern class itself was not so new, having been reflected inthe Schubert calculus on Grassmannians, and the work of the Italian school of algebraicgeometry. On the other hand there was now a framework which produced families ofclasses, whenever there was a vector bundle involved.The prime mechanism then appeared to be this: Given a space X carrying a vectorbundle, implied in the homotopy category a mapping from X to a classifying spaceBG, for the relevant linear group G. For the homotopy theory, the relevant informationis carried by compact subgroups such as the orthogonal groups and unitary groups ofG. Once the cohomology H ∗ (BG) was calculated, once and for all, the contravarianceproperty of cohomology meant that characteristic classes for the bundle would be definedin H ∗ (X) in the same dimensions. For example, the Chern class is really one class withgraded components in each even dimension.This is still the classic explanation, though in a given geometric theory it is profitableto take extra structure into account. When cohomology became ‘extra–ordinary’ withthe arrival of K−theory and Thom’s cobordism theory from 1955 onwards, it was reallyonly necessary to change the letter H everywhere to say what the characteristic classeswere.Characteristic classes were later found for foliations of manifolds; they have (in amodified sense, for foliations with some allowed singularities) a classifying space theoryin homotopy theory.