Relating Enrique surface with K3 and Kummer through involutions and double covers over finite automorphisms on Topological Euler–Poincaré characteristics over complex K3 with Kähler equivalence

Abstract

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.

Homotopy Group of Spheres, Hopf Fibrations and Vil-Larceau Circles

“…There exists 3 regimes depending upon the fact whether , n or that I will summarize below [9][10][11][12][13][14][15][16][17][18][19] ; [1] The trivial homotopy group exists in the mapping of n where the mapping maps all of to single points of which can also be termed as continuously deformable in terms of the mapped surface. [2] For, there exists a degree by which it can be determined that, how many times a sphere is wrapped around itself in the form of the mapping.…”

“…These maps established under equivalence classes keeping the base point fixed, where a continuous map has been made called as null homotopic . These classes of maps becomes an 'equator pinch' where one maps the equator in the form of a pointed sphere (here circle) to a point whose both sides are the upper and lower spheres making it look like a 'bouquet of spheres, where the upper and lowe sphere's pointed equator in the middle makes a pinch and completes the map which formulizes as [14][15][16][17][18][19][20] ;…”

Non-commutativity Over Canonical Suspension η for Genus g ≥1 in Hypercomplex Structures for Potential ρϕ

“…Suspension over a topological space can induce its inert geometry making it bend or create genus which imparts a relation that the same space can be transformed into an algebraic category if we make a grid over that space and in each intersection of the aforesaid grid, the manifolds that are existent in hypercomplex form can be changed from their one structure to the other resulting a change in the number of genera thereby inducing a different category of alterations. But this can only suffice if the suspension functor can be grouped with another suspension functor that opposes its own kind as while creates genus, removes or destroys the genus provided there exists two relations [1,10,11,13],…”

“…will always act on with being the number of genus where 0 and 1 in denotes the suspension or not through the resultant orbit being open or closed making the order in a way as to represent the previously said affine parameter closed through for for an infinite hypercomplex manifold present in topological space [7,8,19,20] that would act in 2 ways via disjoint union among hypercomplex with the other for with represented via and with represented via with and associated with simply denotes the two different hypercomplex without arising any confusion will act in the geometry alteration of genus through the resemblance of (trivial just to denote the case) but (non-trivial in respect of operations at Planck's scale) if acts on , i.e., and acts on , i.e., then this suffice for two such hypercomplex structures where over a value of closed for infinite hypercomplex manifold [6,11,13] representing globally non-commutativity for factors in matrix multiplications [5].…”

“…Considering a ring of where where each element denotes an orbit taking over the ring for an operator to move from the ring to the opposite of ring to establish a factor which corresponds to the permutations upto for the element in set making possible relations over a commutative giving the relation by sufficing relations to be considered below in stepwise formulations [10,11,13].…”