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

Abstract

In this paper, we give a new topological invariant and a kind of characterization to homeomorphisms. We introduce this topological invariant by constructing an algebraic kind of groupoid structure to each topological space, which is an extension of the concept of fundamental groups. This construction we call it “Core fundamental groupoid” and it is different from the fundamental groupoid as in the book and articles of Ronald Brown [30, 31]. Moreover, both groupoid notions are not equivalent, but, it will be a wide subgroupoid of the fundamental groupoid in category-theoretic. In the entire paper, we consider groupoid in the algebraic sense. First, we discuss basic properties of the Core fundamental groupoid and their significant importance in topological invariants. Further, we separately present the Core fundamental groupoid as an algebraic structure and a topo-algebraic structure and investigate their properties separately. We have an explicit description of both the algebraic structure of the groupoid and a unique topological structure of Core fundamental groupoid. Besides, we give a kind of characterization of homeomorphisms in terms of an invariant that we have obtained, i.e., “f : M → N is a homeomorphism if and only if f# 1M → 1N is a topological groupoid isomorphism”. This result addresses the open problem stated in [1, 19, 24, 26] and also gives a kind of characterization for homeomorphic spaces, but, computationally it seems difficult and it becomes sometimes trivial when intact with topology. Induced groupoid homomorphism, induced base map, topological and smooth structure on the Core fundamental groupoid, homotopic properties of induced groupoid homomorphisms of continuous maps and topological groupoid homomorphism-related properties of induced groupoid homomorphisms are discussed. We also present the relation of homotopy type on Core fundamental groupoid.