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.

From symmetric product CFTs to AdS3

Correlators in symmetric orbifold CFTs are given by a finite sum of admissible branched covers of the 2d spacetime. We consider a Gross-Mende like limit where all operators have large twist, and show that the corresponding branched covers can be described via a Penner-like matrix model. The limiting branched covers are given in terms of the spectral curve for this matrix model, which remarkably turns out to be directly related to the Strebel quadratic differential on the covering space. Interpreting the covering space as the world-sheet of the dual string theory, the spacetime CFT correlator thus has the form of an integral over the entire world-sheet moduli space weighted with a Nambu-Goto-like action. Quite strikingly, at leading order this action can also be written as the absolute value of the Schwarzian of the covering map.Given the equivalence of the symmetric product CFT to tensionless string theory on AdS3, this provides an explicit realisation of the underlying mechanism of gauge-string duality originally proposed in [1] and further refined in [2].

The Most Refined Axiom for a Digital Covering Space and Its Utilities

This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local (k0,k1)-isomorphisms and use the most simplified local (k0,k1)-isomorphism. This approach is indeed a key step to make the axioms for a digital covering space very refined. In this paper, the most refined local (k0,k1)-isomorphism is proved to be a (k0,k1)-covering map, which implies that the earlier axioms for a digital covering space are significantly simplified with one axiom. This finding facilitates the calculations of digital fundamental groups of digital images using the unique lifting property and the homotopy lifting theorem. In addition, consider a simple closed k:=k(t,n)-curve with five elements in Zn, denoted by SCkn,5. After introducing the notion of digital topological imbedding, we investigate some properties of SCkn,5, where k:=k(t,n),3≤t≤n. Since SCkn,5 is the minimal and simple closed k-curve with odd elements in Zn which is not k-contractible, we strongly study some properties of it associated with generalized digital wedges from the viewpoint of fixed point theory. Finally, after introducing the notion of generalized digital wedge, we further address some issues which remain open. The present paper only deals with k-connected digital images.

The Birman-Hilden property of covering spaces of nonorientable surfaces

UDC 517.5 Let p : N ˜ → N be a finite covering space of nonorientable surfaces, where χ ( N ˜ ) < 0 . We search whether or notphasthe Birman – Hilden property.

Calibrations for minimal networks in a covering space setting

In this paper, we define a notion of calibration for an approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover, we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon.

Classification of ribbon 2-knots presented by virtual arcs with up to four crossings

We consider classification of the oriented ribbon 2-knots presented by virtual arcs with up to four crossings. We show the difference by the 2-fold branched covering space, the Alexander polynomial, the number of representations of the knot group to SL[Formula: see text], [Formula: see text] a finite field, and the twisted Alexander polynomial.

Modeling Spacetime as a Branched Covering Space over 2-Knots

In this paper we review a proposal to represent the geometric degrees of freedom of the gravitational field as a branched covering space, and introduce a new application of this in which the branch loci are 1- or 2-knots. This allows one to construct arbitrary smooth, closed 3- and 4-manifolds with enough geometric and topological information to write down a partition function and calculate statistical quantities in the thermodynamic limit. Further, we find clear evidence for a dimensional reduction of the spacetime geometry from four to two. As an example, we choose a family of smooth 4-manifolds presented in this way, and calculate the entropy of the system.

A Remark on Configuration Spaces of Two Points

We prove a homotopy invariance result for a certain covering space of the space of ordered configurations of two points in M × X where M is a closed smooth manifold and X is any fixed aspherical space which is not a point.