Fundamental groupoid and Whisker topology

Authors
Ali Pakdaman
Freshteh Shahini
Abstract
Introduction

Algebraic structures on topological spaces can’t distinguish all the non-homeomorphic spaces. Recently, by equipping some of these algebraic structures with topology, one can separate the non-homeomorphic spaces with the same algebraic structures. Particularly, the fundamental group is equipped by the various types of topologies which some of them make it a topological group and some of them don't‎. Virk‎ ‎and‎ ‎Zastro‎w has‎ ‎studied‎ ‎the‎ ‎generalizations‎ ‎of‎ ‎the‎ ‎existing‎‎‎ ‎topologies‎ ‎on‎ ‎the‎ ‎fundamental‎‎group‎ ‎to‎ ‎the‎ ‎universal‎ ‎path‎ ‎space‎ with a proper comparison. If we remove the condition α(0) = x0 from the definition of the universal path space, we obtain an object that have been discussed in mathematical literatures under the name “ fundamental groupoid”. Indeed, fundamental groupoid denoted by πX , is the category of homotopy classes of paths in X as the morphisms and has the set X as the objects set. For any x , y ∈X , the set πXx,y is the set of homotopy classes of paths in X from x to y . We can consider the object group at x , πXx , as the well-known fundamental group π1X,x. ‎

‎ In order to answer the question how these topologies can be generalized on the fundamental groupoids, the authors has introduced the Lasso topology on the fundamental groupoid of a locally path connected space in which makes it a topological groupoid. A topological groupoid is a groupoid G together with topologies on G and G0 such that the structure maps are continuous‎.

R‎. ‎Brown and G‎. ‎Danesh-Naruie were the first and only ones to take this step‎. ‎They have defined a topology on a quotient of the fundamental groupoid such that it became a topological groupoid when the given space X is locally path connected and semilocally simply connected.

Here, we introduce whisker topology on the fundamental groupoid of a locally path connected space X in which it’s basis is known and by some assumptions, we can consider it as a generalization of the whisker topology on the fundamental group.

Material and methods

‎‎ For a given topological space (X,τ) , let [α]∈πX(x,y) ‎, where x,y∈X . ‎If V , ‎ W are open neighborhoods of x ‎, y, respectively‎, one can define

Nα,V,W:=β∈πX β≃γ*α*λ, γI⊆V, λI⊆W} ,

where γ1=α0=x and α1=λ0=y.

Theorem: ‎The family‎

‎Nα,V,W;α∈πXx,y,x∈V∈τ,y∈W∈τ

‎forms a basis for a topology on fundamental groupoid‎.

‎The topology that is generated by this basis‎, ‎is called Whisker topology.

Conclusion

Based on the results that we presented in this paper:

If X is a small loop transfer space;

The multiplication map m: πwhX × πwhX → πwhX is continuous.

The fundamental groupoid with the Whisker topology is a topological groupoid.

The inherited topology from fundamental groupoid πwhX on the object group πwhX(x) equals to the Whisker topology on π1X,x.
Keywords
Fundamental groupoid
Whisker topology
Topological groupoid
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