Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 495orthogonally along the common core circle of both bands, which is also thecore circle of the solid torus.The complex version of ˜KO(X), called ˜K(X), is constructed in the sameway as ˜KO(X) but using vector bundles whose fibers are vector spaces overC rather than R. The complex form of Bott Periodicity asserts that ˜K(S n )is Z for n even and 0 for n odd, so the period is two rather than eight.The groups ˜K(X) and ˜KO(X) for varying X share certain formal propertieswith the cohomology groups studied in classical algebraic topology[Hatcher (2002)]. Using a more general form of Bott periodicity, it is infact possible to extend the groups ˜K(X) and ˜KO(X) to a full cohomologytheory, families of Abelian groups ˜K n (X) and ˜KO n (X) for n ∈ Z thatare periodic in n of period two and eight, respectively. However, thereis more algebraic structure here than just the additive group structure.Namely, tensor products of vector spaces give rise to tensor products ofvector bundles, which in turn give product operations in both real andcomplex K−theory similar to cup product in ordinary cohomology. Furthermore,exterior powers of vector spaces give natural operations withinK−theory (for more development, see next section, below).4.3.1 The Second Vector Bundle of the Manifold MLet (E, π, M) be a vector bundle over the biodynamical manifold M withfibre addition + E : E × M E → E and fibre scalar multiplication m E t : E →E. Then (T E, π E , E), the tangent bundle of the manifold E, is itself avector bundle, with fibre addition denoted by + T E and scalar multiplicationdenoted by m T tE . The second vector bundle structure on (T E, T π, T M), isthe ‘derivative’ of the original one on (E, π, M). In particular, the space{Ξ ∈ T E : T π.Ξ = 0 ∈ T M} = (T p) −1 (0) is denoted by V E and is calledthe vertical bundle over E. Its main characteristics are vertical lift andvertical projection (see [Kolar et al. (1993)] for details).All of this is valid for the second tangent bundle T 2 M = T T M of amanifold, but here we have one more natural structure at our disposal.The canonical flip or involution κ M : T 2 M → T 2 M is defined locally by(T 2 φ ◦ κ M ◦ T 2 φ −1 )(x, ξ; η, ζ) = (x, η; ξ, ζ).where (U, φ) is a local chart on M (this definition is invariant under changesof charts). The flip κ M has the following properties (see [Kolar et al.(1993)]):