Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 511theory for manifolds, in particular; and numerous other connections withclassical algebraic problems were found. A little later a branch of the theoryfor operator algebras was fruitfully developed. It also became clear thatK−theory could play a role in algebraic cycle theory in algebraic geometry:here the higher K−groups become connected with the higher codimensionphenomena, which are exactly those that are harder to access. The problemwas that the definitions were lacking (or, too many and not obviouslyconsistent). A definition of K 2 for fields by John Milnor, for example, gavean attractive theory that was too limited in scope, constructed as a quotientof the multiplicative group of the field tensored with itself, with someexplicit relations imposed; and closely connected with central extensions[Milnor and Stasheff (1974)].Eventually the foundational difficulties were resolved (leaving a deepand difficult theory), by a definition of D. Quillen:K n (R) = π n (BGL(R) + ).This is a very compressed piece of abstract mathematics. Here π n is an nthhomotopy group, GL(R) is the direct limit of the general linear groups overR for the size of the matrix tending to infinity, B is the classifying spaceconstruction of homotopy theory, and the + is Quillen’s plus construction.4.5.3 Chern Classes and Chern CharacterAn important properties in K–theory are the Chern classes and Cherncharacter [Chern (1946)]. The Chern classes are a particular type of characteristicclasses (topological invariants, see [Milnor and Stasheff (1974)]).associated to complex vector bundles of a smooth manifold. Recall thata characteristic class is a way of associating to each principal bundle on atopological space X a cohomology class of X. The cohomology class measuresthe extent to which the bundle is ‘twisted’ – particularly, whether itpossesses sections or not. In other words, characteristic classes are globalinvariants which measure the deviation of a local product structure from aglobal product structure. They are one of the unifying geometric conceptsin algebraic topology, differential geometry and algebraic geometry. 77 Recall that characteristic classes are in an essential way phenomena of cohomologytheory – they are contravariant functors, in the way that a section is a kind of functionon a space, and to lead to a contradiction from the existence of a section we do needthat variance. In fact cohomology theory grew up after homology and homotopy theory,which are both covariant theories based on mapping into a space; and characteristicclass theory in its infancy in the 1930s (as part of obstruction theory) was one major