A segregated approach is used through several maps and classifiers where a study has been conducted to establish a concrete relational equivalence among several hypercomplex structures over three—order categorization tables namely ‘Enrique surface classification’ and ‘Enrique-K3-Complex(K3)-Kähler-Kummer surface characteristics’ with ‘K3– Complex(K3)-Kähler-Kummer(K3)-Kummer classifiers’ taking into account several notions of algebraic topology and algebraic geometry. This paper is the continuation of the previous preprint https://doi.org/10.21203/rs.3.rs-1635957/v1. For the Enrique 7—classifiers are considered namely I, II, III, IV, V, VI, VII.

<p>Representing the β bundles as an infinite fibre that when acts on the geometries been presented as -1 for hyperbolic or saddle curvature and +1 for elliptic or positive curvature with ±1 for both types of curvatures with the exception of 0 curvature Euclidean geometry can cause deformation or suspension η in the complex topological space T^* thereby creating genus g≥1 or with the  R_∩ formalism making the entire manifold M in the form of a suspended disc ∑D_Λ  to collapse as a ring M_R for non–intersection of infinite loops ∮_∞γ as the trivial assumptions for the extreme degrees of freedom.</p>

<p>Type-II emphasizing Type-II(B) in Ramond–Ramond Sector has been analyzed and computed from the Atiyah–Hirzebruch spectral sequence taking E_i sheets for the concerned values of i = 4=∞ and for E_n^(p,q) for n=1,2,3; 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.</p>

<p>This paper contributes an extensive analysis of Dark Matter (DM) along with associated materials regarding the interpretation of its evidence from 'cosmic microwave background radiation’, ‘gravitational lensing’, ‘galactic mergers and collisions’, ‘evolution and formation of galaxies’, ‘galactic clusters’, ‘Lambda-CDM models of the observable universe. Massive Astrophysical Compact Halo Objects (MACHOs), Weakly Interacting Massive Particles (WIMP’s), Strongly Interacting Massive Particles (SIMPs), along with hot, warm, and cold DM, shedding an extensive overview of the γ_elastic of the cold DM after the chemical equilibrium is reached and kinetic decoupling happens earlier for γ_(in-elastic) accounting for 85% of the dominated matter in the universe. Scaler-Vector-Tensor (SVT) Gravity, MOdified Newtonian Dynamics (MONDs), and Gravity of Entropy (GoE) have emerged as a supplement of the modified General Relativity (GR) as a source of non-baryonic DM scenarios in the universe. Observable evidence has been taken into account for the ‘dispersion of velocity’, ‘galactic rotation curves’, ‘Lyman-α forest simulations’, ’21 cm red-shift spectra’, ‘satellite sky surveys’, ‘formation of structures’, and ‘supernova measurements of Type Ia’. Direct evidence in the underground chamber for ‘nuclear giggles’ either ‘Cryogenic experiments CRESST, CDMS’ or ‘noble liquid gases XENON’ as a result of DM scattering via VECTOR-PORTALS and Indirect evidence as a result of sky-surveys like satellite telescopes ANTARES, AMANDA or through the ‘collider experiments’ in LHC, CERN, Geneva, Switzerland.</p>

<p>This paper contributes an extensive analysis of Dark Matter (DM) along with associated materials regarding the interpretation of its evidence from 'cosmic microwave background radiation’, ‘gravitational lensing’, ‘galactic mergers and collisions’, ‘evolution and formation of galaxies’, ‘galactic clusters’, ‘Lambda-CDM models of the observable universe. Massive Astrophysical Compact Halo Objects (MACHOs), Weakly Interacting Massive Particles (WIMP’s), Strongly Interacting Massive Particles (SIMPs), along with hot, warm, and cold DM, shedding an extensive overview of the γ_elastic of the cold DM after the chemical equilibrium is reached and kinetic decoupling happens earlier for γ_(in-elastic) accounting for 85% of the dominated matter in the universe. Scaler-Vector-Tensor (SVT) Gravity, MOdified Newtonian Dynamics (MONDs), and Gravity of Entropy (GoE) have emerged as a supplement of the modified General Relativity (GR) as a source of non-baryonic DM scenarios in the universe. Observable evidence has been taken into account for the ‘dispersion of velocity’, ‘galactic rotation curves’, ‘Lyman-α forest simulations’, ’21 cm red-shift spectra’, ‘satellite sky surveys’, ‘formation of structures’, and ‘supernova measurements of Type Ia’. Direct evidence in the underground chamber for ‘nuclear giggles’ either ‘Cryogenic experiments CRESST, CDMS’ or ‘noble liquid gases XENON’ as a result of DM scattering via VECTOR-PORTALS and Indirect evidence as a result of sky-surveys like satellite telescopes ANTARES, AMANDA or through the ‘collider experiments’ in LHC, CERN, Geneva, Switzerland.</p>