KK Theory and K Theory for Type II Strings Formalism

Type-II emphasizing Type-II(B) in Ramond–Ramond Sector  has been analysed and computed from the Atiyah–Hirzebruch spectral sequence taking  sheets for the concerned values of and for for ; several varieties of K–Theories where a transitive approach has been shown from the KK–Theory to K–Theory to String Theory concerning Fredholm modules of Atiyah–Singer Index Theorem and the Baum–Connes conjecture with respect to the Hilbert–A, Hilbert–B  module and c*-algebras also in the reduced form taking Morita equivalence and the Kasparov composition product where extended relations has been provided between the equivalence of noncommutative geometry and noncommutative topology channelized through Poincaré Duality, Thom Isomorphism and Todd class.