Ivancevic_Applied-Diff-Geom

508 Applied Differential Geometry: A Modern IntroductionUsingL = 1 2ẋiẋ i − X i ẋ i + f, H = 1 2ẋiẋ i − f, (i, j, h = 1, 2, 3),g ij = (H + f)δ ij , N j i = −F j i = −δ ih F jh , F ij = ∂X j∂x i− ∂X i∂x j,the solutions of the above differential system are horizontal pregeodesics ofthe Riemann–Jacobi–Lagrangian manifold (R 3 \ E, g ij , N i j ), where E isthe set of equilibrium points, which is included in the surface of equationsin x 1 sin x 2 sin x 3 + cos x 1 cos x 2 cos x 3 = 0.4.5 K−Theory and Its ApplicationsRecall from [Dieudonne (1988)] that the 1930s were the decade of the developmentof the cohomology theory, as several research directions grew togetherand the de Rham cohomology, that was implicit in Poincaré’s work,became the subject of definite theorems. The development of algebraictopology from 1940 to 1960 was very rapid, and the role of homology theorywas often as ‘baseline’ theory, easy to compute and in terms of whichtopologists sought to calculate with other functors. The axiomatization ofhomology theory by Eilenberg and Steenrod (celebrated Eilenberg–SteenrodAxioms) revealed that what various candidate homology theories had incommon was, roughly speaking, some exact sequences (in particular, theMayer–Vietoris Theorem and the Dimension Axiom that calculated thehomology of the point).4.5.1 Topological K−TheoryNow, K–theory is an extraordinary cohomology theory, which consists oftopological K−theory and algebraic K−theory. The topological K–theorywas founded to study vector bundles on general topological spaces, bymeans of ideas now recognisee as (general) K−theory that were introducedby Alexander Grothendieck. The early work on topological K−theory wasdue to Michael Atiyah and Friedrich Hirzebruch.Let X be a compact Hausdorff space and k = R or k = C. Then K k (X)is the Grothendieck group of the commutative monoid 2 which elements are2 Recall that a monoid is an algebraic structure with a single, associative binaryoperation and an identity element; a monoid whose operation is commutative is called acommutative monoid (or, an Abelian monoid); e.g., every group is a monoid and every