Topological Deformations of Manifolds by Algebraic Compositions in Polynomial Rings

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manif...

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Bibliographic Details
Published in:Symmetry (Basel) Vol. 16; no. 5; p. 556
Main Author: Bagchi, Susmit
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.05.2024
MDPI
Subjects:
Algebra
Algebraic Geometry
Composition
Deformation
Equivalence
General Topology
Isomorphism
Manifolds (mathematics)
Mathematical analysis
Mathematics
Numerical analysis
Polynomials
Rings (mathematics)
Simulation methods
Topological manifolds
Topology
manifolds
algebraic sets
topology
polynomial ring
ISSN:2073-8994, 2073-8994
Online Access:Get full text
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Summary:The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism.
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ISSN:2073-8994
2073-8994
DOI:10.3390/sym16050556