A kind of characterization of homeomorphism and homeomorphic spaces by Core fundamental groupoid: a good invariant

• Badiger, Chidanand ; Venkatesh, T. ^[1]
  1. [1] Rani Channamma University.
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 42, Nº. 2, 2023, págs. 273-302
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-4533
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we give new topological invariants and a complete characterization to homeomorphisms. The finding a sufficient condition for homeomorphism and classifying topological spaces up to homeomorphism is the open problemin topology [1, 9, 14]. In this article, the main results are Propositions 3.24, 4.40 and 4.41, and Propositions 4.40 is about complete characterization of homeomorphisms i.e. "f : M -->N is a homeomorphism if and only if f# : pi1M --> pi1N is a groupoid iso-homeomorphism". this is the answer to the open problem [1, 9, 14] mentioned. First, we characterize the homeomorphisms completely. In addition, we resolve the open issue [1, 9, 14] of finding sufficient conditions for two topological spaces to be homeomorphic by giving an invariant. The entire result will be obtained by constructing a new notion, that is an extension of fundamental groups; which is already a topological invariant but not a sufficient one. We extend new theory by defining an algebraic sense of fundamental groupoid by establishing such algebraic structure and a unique topology on it. This fundamental groupoid is different from the fundamental groupoid in [16] and also these two different groupoids (one is algebraic sense and another is category theoretic) are not equivalent. We have an explicit description for algebraic structure groupoid and a unique topological structure on fundamental groupoid. And also we will discuss their topological properties also possibility of smooth structures.

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