A non-Archimedean analogue of Teichmüller space and its tropicalization

In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller space$$\overline{{{\mathcal {T}}}}_g$$ T ¯ g whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the tropical Teichmüller space introduced by Chan–Melo–Viviani as a simplicial completion of Culler–Vogtmann Outer space. As a consequence, Outer space turns out to be a strong deformation retract of the locus of smooth Mumford curves in $$\overline{{\mathcal {T}}}_g$$ T ¯ g .

The minimum deformation retract on the wormhole spacetime

In this paper, we investigate the topology of a wormhole from the viewpoint of the theory of retracting and var-folding. We deduce the equatorial geodesics on the line element of the wormhole and discuss the minimum deformation retract related to this space. Using the Lagrangian equations we find that there is a type of minimum retraction of a wormhole with associated topology. We also extend the result to the [Formula: see text]-dimensional wormhole and show that the end limit of var-folding is 0-dimensional wormhole and obtain the relation between limit var-folding and limit retraction. We find a new application in geometric topology and astrophysics.

Digital co-Hopf spaces

In this work, we deal with co-Hopf space structure of digital images. We prove that a pointed digital image having the same digital homotopy type as a digital co-Hopf space is itself a digital co-Hopf space. We conclude that a k-deformation retract of a digital co-Hopf space is a digital co-Hopf space. We also show that the digital equivalences are digital co-Hopf homomorphisms.

Existence of deformations and dimensional reduction in wormholes and black holes

The deformation retract is, by definition, a homotopy between a retraction and the identity map. We show that applying this topological concept to Ricci-flat wormholes/black holes implies that such objects can get deformed and reduced to lower dimensions. The homotopy theory can provide a rigorous proof to the existence of black holes/wormholes deformations and explain the topological origin. The current work discusses such possible deformations and dimensional reductions from a global topological point of view, it also represents a new application of the homotopy theory and deformation retract in astrophysics and quantum gravity.

On the Deformation Retract of Eguchi-Hanson Space and Its Folding

We introduce the deformation retract of the Eguchi-Hanson space using Lagrangian equations. The retraction of this space into itself and into geodesics has been presented. The deformation retract of the Eguchi-Hanson space into itself and after the isometric folding has been discussed. Theorems concerning these relations have been deduced.

On the Deformation Retract of Kerr Spacetime and Its Folding

The deformation retract of the Kerr spacetime is introduced using Lagrangian equations. The equatorial geodesics of the Kerr space have been discussed. The retraction of this space into itself and into geodesics has been presented. The deformation retract of this space into itself and after the isometric folding has been discussed. Theorems concerning these relations have been deduced.

Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.