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

Abstract

Unlike geometry, spheres in topology have been seen as topological invariants, where their structures are defined as topological spaces. Forgetting, the exact notion of geometry, and the impossibility of embedding one into other, the homotopy relates how one sphere of i dimensions can wrap another sphere of n dimensions.

Establishing Equivariant Class [O] for Hyperbolic Groups

“…Ball having 3. For scalar potential with Riemann there exists [12], a. b. Equivalence class for the isotopy, diffeomorphism, and holomorphism such that for closed interval Teichmüller gives [13],…”

Handle decomposition of a wormhole through Heegaard Splitting from 3 + 1 Space-Time Dimensions

“…And if anyone say that a universe has a genus then this will be terribly horrific, but for the sake of mathematical construction and homeomorphism between a genus 1 wormhole and genus 1 universe -there can there's way out to detach the wormhole from the universe and that is through the Heegaard splitting. Thus, this satisfises homeomorphism where if we say that the universe being a boundary of a topological manifold described as 𝑈 Σ which is a subset of a space-time 𝑈 Σ 1 and wormhole detached from that space-time 𝑈 Σ 2 on a specific patch of cosmos 𝜌 for homeomorphism ℎ on a compact scale -this can be easily monitored, via the equations [1,3] ,…”

“…Introduction -Considering the nature of spatial dimensions of our universe, as it is a 3-manifold or a 3dimensional Euclidean spaces, the main objective of this study is to detach a wormhole from a particular spacetime which it captures making the two separate by using the topological foundations of Heegaard splitting only when it is assumed that the wormholes purely a mathematical structure with a genus of both Σ 1 and Σ 2 to be 1 [1,4] . Construction -It is almost impossible to determine the shape of the universe when there's still a proportion of unobservable universe hidden from us to be observable after time being.…”

“…Structure -To structure the final formulations its non-trivial to denote a three-dimensional torus where it's described as the cartesian product being homoeomorphic to three circles with each circle denoted as 𝑆 1 . Thus, the three torus is defined by [1] , 𝕋 𝑛 = (𝑆 1 ) 𝑛 ∃𝑛 = 3…”

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 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. For the constructions of 𝐾𝐾 − 𝑇ℎ𝑒𝑜𝑟𝑦 ; Morita equivalence is an important tool to 𝑐 *− algebras where for the inequality on the two modules 𝐴 𝑎𝑛𝑑 𝐵 ; for the moulder form 𝐸 on 𝐴 and 𝐵 for the moulder form 𝐸 on 𝐴 and 𝐸 • on 𝐵 (as appeared later in the paper) a homotopy invariant bifunctor can make a Morita equivalence for the 𝐾𝐾 − 𝑇ℎ𝑒𝑜𝑟𝑦 through 𝐾𝐾(𝐴, 𝐵) 𝑎𝑛𝑑 𝐾𝐾(𝐵, 𝐶) for 𝐴, 𝐵, 𝐶 as 𝑐 *− algebras; there's for the modular form 𝐸 having elements 𝜀, 𝜖 the inequality represents the form < 𝜀, 𝜖 >< 𝜖, 𝜀 >≤||< 𝜀, 𝜀 > ||< 𝜖, 𝜖 > where for the 𝐴 − 𝑚𝑜𝑑𝑢𝑙𝑒 ; the above relation holds and taking the 𝐵 − 𝑚𝑜𝑑𝑢𝑙𝑒 representing the 𝑐 *− algebraic pair 𝐾𝐾(𝐴, 𝐵) 𝑎𝑛𝑑 𝐾𝐾(𝐵, 𝐶) where one finds the combined form over the composition product representing 𝐾𝐾(𝐴, 𝐶) and the Morita equivalence to be represented in a specific formulation as to be proved throughout the paper [12,13] .…”