Research ArticleOpen AccessGoogle Scholar indexed
The Homotopical Proof of Π<sub>1</sub> (<i>S</i>, <i>x<sub>o</sub></i>) as a Fundamental Group in a General Interval
Department of Mathematical Sciences, University of Mines and Technology, Tarkwa, Ghana
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana
Department of Mathematical Sciences, University of Mines and Technology, Tarkwa, Ghana
Department of Mathematics and Statistics, University of Energy and Natural Resources, Sunyani, Ghana
- 1 Department of Mathematical Sciences, University of Mines and Technology, Tarkwa, Ghana
- 2 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana
- 3 Department of Mathematical Sciences, University of Mines and Technology, Tarkwa, Ghana
- 4 Department of Mathematics and Statistics, University of Energy and Natural Resources, Sunyani, Ghana
Advances in Pure Mathematics·Volume 11 (2021)·Pages 377–385·Published 17 May 2021·DOI10.4236/apm.2021.115024
Copy link · social · email
Abstract
The aim of this study is to establish that, the equivalent class which is made up of homotopic loops is a group with respect to in the general interval [ m , n ] . The study proved from homotopical point of view that is associative, has an identity and inverse function. The study established with proof that is a fundamental group in [ m , n ] , .
KeywordsHomotopyFundamental GroupHomeomorphismEquivalent ClassPath Concatenation
- Armstrong, M.A. (1983) Basic Topology. Springer-Verlag, New York, 87-117. https://doi.org/10.1007/978-1-4757-1793-8
- Bredon, G.E. (1997) Topology and Geometry. Springer-Verlag, New York.
- Brown, R. (2018) Non Abelian Algebraic Topology. 1-2.
- Arnold, B.H. (1949) A Topological Proof of the Fundamental Theorem of Algebra. The American Mathematical Monthly, 56, 465-466. https://doi.org/10.2307/2305130
- Sims, B.T. (1976) Fundamentals of Topology. Macmillan Publishing Co. Inc., New York, 135-142.
- Eda, K. and Kawamura, K. (1998) The Fundamental Groups of One-Dimensional Spaces. Topology and Its Applications, 87, 163-172. https://doi.org/10.1016/S0166-8641(97)00167-3
- Gamelin, T.W. and Greene, R.E. (1999) Introduction to Topology. Courier. 67.
- Cannon, J.W. and Conner, G.R. (2006) On the Fundamental Group of One Dimensional Spaces. Topology and Its Applications, 153, 2648-2655. https://doi.org/10.1016/j.topol.2005.10.008
- Dodson, C.T.J. and Parker, P.E. (1997) A User’s Guide to Algebraic Topology. Kluwer Academic Publishers, Dordrecht, 120-122. https://doi.org/10.1007/978-1-4615-6309-9
- Munkres, J.R. (2000) Topology. Featured Titles for Topology Series. Prentice Hall Incorporated, Hoboken.
- Brew, L., Obeng-Denteh, W. and Zigli, D.D. (2019) Application of Homotopy to the Ageing Process of Human Body within the Framework of Algebraic Topology. Journal of Mathematics Research, 11, 21-25. https://doi.org/10.5539/jmr.v11n4p21
- Zigli, D.D., Brew, L. and Otoo, H. (2020) Homotopical Proof of Π 1 ( S , x o )as a Fundamental Group with respect to “。” in the Interval [0, n ]. International Journal of Algebra, 14, 191-201. https://doi.org/10.12988/ija.2020.91253
- Greenberg, M.J. and Harper, J.R. (1981) Algebraic Topology: A First Course. Addison Wesley, Boston, 6-31.
- Zigli, D.D., Obeng-Denteh, W. and Owusu-Mensah, I. (2017) On the Candid Appraisal of the Proof of Π 1 ( S , x o ) as a Fundamental Group with Respect to “。”. International Journal of Advances in Mathematics, 2017, 20-26.
- Hocking, J.G. and Young, G.S. (1961) Topology. Addison-Wesley publishing Co. Inc., Reading.