Topological Properties of Braid-Paths Connected 2-Simplices in Covering Spaces under Cyclic Orientations

In general, the braid structures in a topological space can be classified into algebraic forms and geometric forms. This paper investigates the properties of a braid structure involving 2-simplices and a set of directed braid-paths in view of algebraic as well as geometric topology. The 2-simplices are of the cyclically oriented variety embedded within the disjoint topological covering subspaces where the finite braid-paths are twisted as well as directed. It is shown that the generated homotopic simplicial braids form Abelian groups and the twisted braid-paths successfully admit several varieties of twisted discrete path-homotopy equivalence classes, establishing a set of simplicial fibers. Furthermore, a set of discrete-loop fundamental groups are generated in the covering spaces where the appropriate weight assignments generate multiplicative group structures under a variety of homological formal sums. Interestingly, the resulting smallest non-trivial group is not necessarily unique. The proposed variety of homological formal sum exhibits a loop absorption property if the homotopy path-products are non-commutative. It is considered that the topological covering subspaces are simply connected under embeddings with local homeomorphism maintaining generality.

Interactions between Homotopy and Topological Groups in Covering (C, R) Space Embeddings

The interactions between topological covering spaces, homotopy and group structures in a fibered space exhibit an array of interesting properties. This paper proposes the formulation of finite covering space components of compact Lindelof variety in topological (C, R) spaces. The covering spaces form a Noetherian structure under topological injective embeddings. The locally path-connected components of covering spaces establish a set of finite topological groups, maintaining group homomorphism. The homeomorphic topological embedding of covering spaces and base space into a fibered non-compact topological (C, R) space generates two classes of fibers based on the location of identity elements of homomorphic groups. A compact general fiber gives rise to the discrete variety of fundamental groups in the embedded covering subspace. The path-homotopy equivalence is admitted by multiple identity fibers if, and only if, the group homomorphism is preserved in homeomorphic topological embeddings. A single identity fiber maintains the path-homotopy equivalence in the discrete fundamental group. If the fiber is an identity-rigid variety, then the fiber-restricted finite and symmetric translations within the embedded covering space successfully admits path-homotopy equivalence involving kernel. The topological projections on a component and formation of 2-simplex in fibered compact covering space embeddings generate a prime order cyclic group. Interestingly, the finite translations of the 2-simplexes in a dense covering subspace assist in determining the simple connectedness of the covering space components, and preserves cyclic group structure.

Sifat-Sifat Morfisma di dalam Kategori Ruang Penutup Ruang Topologis yang Terhubung Lintasan (On the Morphisms of the Category of Path Connected Covering Spaces )

Untuk sebarang ruang topologis $X$ dapat dibentuk $Cov_X$ yaitu kategori \linebreak ruang penutup $X$ yang terhubung lintasan. Pada tulisan ini akan dibahas syarat perlu dan cukup eksistensi morfisma antara dua ruang penutup yang terhubung lintasan lokal. Untuk sebarang $x_0 \in X$ dan grup fundamental $G=\pi_1(X,x_0)$, dapat dibentuk kategori $SetG$, yaitu kategori semua himpunan yang dilengkapi aksi kanan oleh $G$. Selanjutnya dibentuk fungtor $F$ dari $Cov_X$ ke $SetG$. Dalam tulisan dibuktikan bahwa $F$ bersifat \textit{fully faithful} jika $X$ terhubung lintasan dan terhubung lintasan lokal. Akibatnya untuk mengidentifikasi morfisma-morfisma antara dua obyek $A$ dan $B$ di $Cov_X$ dapat dilakukan dengan cara melihat sifat morfisma-morfisma antara $F(A)$ dan $F(B)$. (For any topological space $X$, we can construct the category of path \linebreak connected covering spaces of $X$, denoted by $Cov_X$. In this paper we study a sufficient and necesarry condition for the existence of morphism between two locally path \linebreak connected covering spaces. For every $x_0 \in X$ and fundamental group $G=\pi_1(X,x_0)$, we can construct the category of sets with right action of $G$, denoted by $SetG$. \linebreak Furthermore, we can define a functor $F$ from $Cov_X$ to $SetG$. We proof that the functor $F$ is fully faithul if $X$ is path connected and locally path connected. From this result, we can identify morphisms between $A$ and $B$ in $Cov_X$ by using the properties of morphisms between $F(A)$ and $F(B)$. )