The Homotopical Proof of Π₁(<i>S</i>, <i>x_o</i>) as a Fundamental Group in a General Interval  

作　　者：Zigli David Delali Obeng-Denteh William Brew Lewis Ansah Richard Kwame Zigli David Delali;Obeng-Denteh William;Brew Lewis;Ansah Richard Kwame(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 Mathematics and Statistics, University of Energy and Natural Resources, Sunyani, Ghana)

机构地区：[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 Mathematics and Statistics, University of Energy and Natural Resources, Sunyani, Ghana

出　　处：《Advances in Pure Mathematics》2021年第5期377-385,共9页理论数学进展（英文）

摘　　要：The aim of this study is to establish that, the equivalent class <img src="Edit_d35dd794-39a5-4ce4-992b-5130559b3c82.png" width="70" height="22" alt="" /> which is made up of homotopic loops is a group with respect to <img src="Edit_3577ec7c-e6f5-4d71-8bd5-c63ea8fdb24f.png" width="30" height="15" alt="" /> in the general interval <span style="white-space:nowrap;">[<em>m</em>,<em>n</em>]</span>. The study proved from homotopical point of view that <img src="Edit_4cb511c3-e469-47e3-bd9c-e971594f939c.png" width="70" height="22" alt="" /> is associative, has an identity and inverse function. The study established with proof that <img src="Edit_39497a4b-b0e9-40d9-8f31-49816e760d6a.png" width="70" height="22" alt="" /> is a fundamental group in <span style="white-space:nowrap;">[<em>m</em>,<em>n</em>]</span> ,<img src="Edit_077b19f1-afb3-41f5-8d39-df073165c9dc.png" width="75" height="18" alt="" />.The aim of this study is to establish that, the equivalent class <img src="Edit_d35dd794-39a5-4ce4-992b-5130559b3c82.png" width="70" height="22" alt="" /> which is made up of homotopic loops is a group with respect to <img src="Edit_3577ec7c-e6f5-4d71-8bd5-c63ea8fdb24f.png" width="30" height="15" alt="" /> in the general interval <span style="white-space:nowrap;">[<em>m</em>,<em>n</em>]</span>. The study proved from homotopical point of view that <img src="Edit_4cb511c3-e469-47e3-bd9c-e971594f939c.png" width="70" height="22" alt="" /> is associative, has an identity and inverse function. The study established with proof that <img src="Edit_39497a4b-b0e9-40d9-8f31-49816e760d6a.png" width="70" height="22" alt="" /> is a fundamental group in <span style="white-space:nowrap;">[<em>m</em>,<em>n</em>]</span> ,<img src="Edit_077b19f1-afb3-41f5-8d39-df073165c9dc.png" width="75" height="18" alt="" />.

关 键 词：HOMOTOPY Fundamental Group HOMEOMORPHISM Equivalent Class Path Concatenation 

分 类 号：O17[理学—数学]