Topological fundamental groupoids: Brown's topology

Year 2023, , 896 - 906, 15.08.2023

Ali Pakdaman , Fereshteh Shahini

https://doi.org/10.15672/hujms.1205441

Abstract

In this paper, we generalize the Brown$^{^,}$s topology on the fundamental groupoids. For a locally path connected space $X$ and a totally disconnected normal subgroupoid $M$ of $\pi X$, we define a topology on the quotient groupoid $\dfrac{\pi X}{M}$ which is a generalization of what introduced by Brown for locally path connected and semilocally simply connected spaces. We prove that $\dfrac{\pi X}{M}$ equipped with this topology is a topological groupoid. Also, we will find a class of subgroupoids of topological groupoids whose their related quotient groupoids will be topological groupoids. By using this, we show that our topology on $\dfrac{\pi X}{M}$ is equivalent to the quotient of the Lasso topology on the topological fundamental groupoids, $\dfrac{\pi^L X}{M}$.

Keywords

fundamental groupoid‎, ‎topological groupoid‎, ‎quotient groupoid.

References

• [1] R. Brown, Topology and Groupoids. Deganwy, United Kingdom: BookSurge LLC, 2006.
• [2] R. Brown and G. Danesh-Naruie,Topological Fundamental Groupoid as Topological Groupoid, Proc. Edinburgh Math. Soc. 19, 237–244, 1975.
• [3] R. Brown and J.P.L. Hardy,Topological groupoid: I. Universal constructions, Math. Nachr. 71 (1), 273–286, 1976.
• [4] N. Brodskiy, J. Dydac, B. Labuz, A. Mitra, Covering maps for locally path-connected spaces, Fundam. Math. 218, 13–46, 2012.
• [5] P. J. Higgins, Notes on Categories and Groupoids, Van Nostrand, 1971.
• [6] O. Mucuk, T. Sahan, N. Alemdar, Normality and Quotients in Crossed Modules and Group-groupoids, Appl. Categ. Struct. 23, 415–428, 2015.
• [7] K. MacKenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society Lecture Note Series 124, 1987.
• [8] H. Fischer, D. Repovš, Ž. Virk, A. Zastrow, On semilocally simply connected spaces, Topol. Appl. 158, 397–408, 2011.
• [9] Z. Virk and A. Zastrow,A new topology on the universal path space, Topol. Appl. 231, 186–196, 2017.
• [10] H. Spanier, Algebraic Topology, McGraw-Hill Book Co, New York-Toronto, Ont.- London, 1966.
• [11] A. Pakdaman, F. Shahini, The fundamental groupoid as a topological groupoid: Lasso topology, Topol. Appl. 302, 107837, 2021.