Ivancevic_Applied-Diff-Geom

510 Applied Differential Geometry: A Modern Introduction(1) K(X ×S 2 ) = K(X)⊗K(S 2 ), and K(S 2 ) = [H]/(H −1) 2 , where H isthe class of the tautological bundle on the S 2 = P 1 , i.e., the Riemannsphere as complex projective line;(2) ˜K n+2 (X) = ˜K n (X);(3) Ω 2 BU ≃ BU × Z.In real K−theory there is a similar periodicity, but modulo 8.4.5.2 Algebraic K−TheoryOn the other hand, the so–called algebraic K–theory is an advanced partof homological algebra concerned with defining and applying a sequenceK n (R) of functors from rings to Abelian groups, for n = 0, 1, 2, .... Here,for traditional reasons, the cases of K 0 and K 1 are thought of in somewhatdifferent terms from the higher algebraic K−groups K n for n ≥ 2. Infact K 0 generalizes the construction of the ideal class group, using projectivemodules; and K 1 as applied to a commutative ring is the unit groupconstruction, which was generalized to all rings for the needs of topology(simple homotopy theory) by means of elementary matrix theory. Thereforethe first two cases counted as relatively accessible; while after thatthe theory becomes quite noticeably deeper, and certainly quite hard tocompute (even when R is the ring of integers).Historically, the roots of the theory were in topological K–theory (basedon vector bundle theory); and its motivation the conjecture of Serre 5 thatnow is the Quillen–Suslin Theorem. 6Applications of K−groups were found from 1960 onwards in surgeryin topology because of the connection of their cohomology with characteristic classes,for which all the (unstable) homotopy groups could be calculated. These spaces are the(infinite, or stable) unitary, orthogonal and symplectic groups U, O and Sp.5 Jean–Pierre Serre used the analogy of vector bundles with projective modules tofound in 1959 what became algebraic K−theory. He formulated the Serre’s Conjecture,that projective modules over the ring of polynomials over a field are free modules; thisresisted proof for 20 years.6 The Quillen–Suslin Theorem is a Theorem in abstract algebra about the relationshipbetween free modules and projective modules. Projective modules are modules that arelocally free. Not all projective modules are free, but in the mid–1950s, Jean–Pierre Serrefound evidence that a limited converse might hold. He asked the question: Is everyprojective module over a polynomial ring over a field a free module? A more geometricvariant of this question is whether every algebraic vector bundle on affine space is trivial.This was open until 1976, when Daniel Quillen and Andrei Suslin independently provedthat the answer is yes. Quillen was awarded the Fields Medal in 1978 in part for hisproof.