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

Abstract

Any matrix multiplication is non-commutative which has been shown here in terms of suspension‡, annihilator, and factor as established over a ring following the parameter k over a set of elements upto n for an operator to map the ring R to its opposite Rop having been through a continuous representation of permutation upto n-cycles being satisfied for a factor f along with its inverse f-1 over a denoted orbit γ on k-parameterized ring justified via suspension η ∈ η0, η1 implying the same global non-commutativity for the annihilator A. This will be used for the construction of the genus–alteration scenario where the suspension η0 acting with its opponent η1 on any topological space J can alter the geometry making a change in the manifolds for taking over the Boolean (1,0) satisfying the concerned operations.

Suspension η for β bundles in ±1 geodesics in g≥1 genus creations for loops for a Topological String Theory Formalism

“…Where the genus can be represented by the 𝐸𝑢𝑙𝑒𝑟'𝑠 𝑝𝑜𝑙𝑦ℎ𝑒𝑑𝑟𝑎 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 represented by 𝜒 having the same formalism with the subscript form 𝜒 𝜊 to be reproduced for the genus structure over, 𝜒 = 𝑉𝑒𝑟𝑡𝑖𝑐𝑒𝑠 − 𝐸𝑑𝑔𝑒𝑠 + 𝐹𝑎𝑐𝑒𝑠 = 2 − 2𝑔 ≅ 𝜂 Thus, in any forms for the suspension 𝜂, when we are considering the flowsthere happens a cancellation over the flows taking on the complex topological space denoted as 𝑇 * Where we can create a set for that suspension 𝜂 = {𝜂, 𝜂 × } such that for 𝜂 one can find the genus creation scenario 𝑔 ≥ 1 for the induced weight 𝜛 which will act on the space 𝑇 * in a way to deform the volume of that space 𝑇 𝜈 * in a way to deform over a mapping parameter 𝜁 which will create either of the 2 -𝑝𝑜𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠 for the set element of suspension 𝜂 = {𝜂 + , 𝜂 × } that either will cancelling out creating no genus 𝑔 = 0 or will create a genus 𝑔 ≥ 1 with the relations being shown as [1,2] ,…”

“…occur in positive signatures of the matrix as stated in section [III.II -3.C.I.a] with the increase in dimensions with positive curvatures denoted by Ω +1 Γ while the increment with negative curvatures denoted as Ω −1 Γhowever, there is a case which is extremely non-trivial where the manifold ℳ takes both the positive and negative curvatures and thus (without any particular increment in either positive with the negative remaining the same or increment in either negative with positive remaining the same or in general case both are changing simultaneously as related to the dimensional increment factor 𝜉 ‡, †, ‡ † )the formulation can take for a coherent factor ‡ †, ‡, † for the denoted notation Ω ±1 Γ -3-relations can be stated as [5,6,[1][2][3] ,…”

Knot Theoretic Closure for a Definite Metric in a Finite Time

“…Such that for any twist, there exists two operations; the 'in' operation ↗ ̅ where the manifold bends by entering into the genus ↗ ̅ and the 'out' operation ↗ ̿ ; such that ↗ ̅ , ↗ ̿ exists as a subset of ↗ as ↗ ̅ , ↗ ̿ ⊂ ↗ for the evolution period *𝑇 ↗+ over a metric representation (𝐻, 𝑔) in such a way that there exists a generation of a 'throat' or a 'space arising out of deforming the metric (𝐻, 𝑔)' for a structure formulation of that generating throat having an affine value 𝜕 𝑚,𝑛 where the representation takes place as [3] ,…”

Instability in the Linkage of Topological Spaces Due to Background Ghosts

“…Taking a background space 𝐵 having the boundary 𝜕 for a coherent norm of boundary space 𝜕𝐵 there can be a relation through nodes 𝑁 acting on a stable parameter 𝛾 on the before mentioned background space 𝜕𝐵 which in essence is a complex topological space denoted as 𝑇 ⋆ for a mapping parameter ⋆ such that the nodes or links 𝑁 can take two values for 𝜕𝐵 [1][2][3][4] , # { stable parameter 𝛾 unstable parameter 𝛾 × Denoting # as an affine parameter for the periodicity 𝜌 determining the oscillation factor 𝜀 to give the notions [5][6] ,…”

“…The ghosts which here termed as instabilities arise out of the background oscillations of the topological space 𝑇 * where the instabilities arise out of two factor and affects the third which is the boundary of the geometric structure 𝜕𝐵 which links to each other in a way of dimensions that can be same or cannot such that this 𝜕𝐵 can be formulated via the structure dependent on the geometric spaces having the form [19][20][21][22][23] ,…”