Characteristic Class of Isotopy for Surfaces

Ren, Yuxue; Wen, Chengfeng; Zhen, Shengxian; Lei, Na; Luo, Feng; Gu, Daqing

doi:10.1007/s11424-020-9053-8

Cited by 1 publication

(2 citation statements)

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“…This leads to the definition of the formation of the isotopy class and smooth imbedding, which is stated as follows, considering a family of imbedding [6,7]. Definition 3.…”

Section: Definition 2 If the Injective Function Imentioning

confidence: 99%

“…The isotopy of a locally flat submanifold can be suitably extended under covering maps if the locally flat submanifold is isomorphic to the corresponding proper imbedding and the respective submanifold is compact. Furthermore, two simplicial embeddings are isotopic if there is an isotopic homeomorphism maintaining Haefliger-Wu conditions [7].…”

Section: Definition 2 If the Injective Function Imentioning

confidence: 99%

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Topological Deformations of Manifolds by Algebraic Compositions in Polynomial Rings

Bagchi

2024

Symmetry

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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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“…This leads to the definition of the formation of the isotopy class and smooth imbedding, which is stated as follows, considering a family of imbedding [6,7]. Definition 3.…”

Section: Definition 2 If the Injective Function Imentioning

confidence: 99%

Section: Definition 2 If the Injective Function Imentioning

confidence: 99%