{"title":"空间n(m)的基本群","authors":"A. Paśko","doi":"10.15421/242207","DOIUrl":null,"url":null,"abstract":"In the present paper the spaces $\\Omega_n(m)$ are considered. The spaces $\\Omega_n(m)$, introduced in 2018 by A.M. Pasko and Y.O. Orekhova, are the generalization of the spaces $\\Omega_n$ (the space $\\Omega_n(2)$ coincides with $\\Omega_n$). The investigation of homotopy properties of the spaces $\\Omega_n$ has been started by V.I. Ruban in 1985 and followed by V.A. Koshcheev, A.M. Pasko. In particular V.A. Koshcheev has proved that the spaces $\\Omega_n$ are simply connected. We generalized this result proving that all the spaces $\\Omega_n(m)$ are simply connected. In order to prove the simply connectedness of the space $\\Omega_n(m)$ we consider the 1-skeleton of this space. Using 1-cells we form the closed ways that create the fundamental group of the space $\\Omega_n(m)$. Using 2-cells we show that all these closed ways are equivalent to the trivial way. So the fundamental group of the space $\\Omega_n(m)$ is trivial and the space $\\Omega_n(m)$ is simply connected.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"13 3","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The fundamental group of the space $\\\\Omega_n(m)$\",\"authors\":\"A. Paśko\",\"doi\":\"10.15421/242207\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the present paper the spaces $\\\\Omega_n(m)$ are considered. The spaces $\\\\Omega_n(m)$, introduced in 2018 by A.M. Pasko and Y.O. Orekhova, are the generalization of the spaces $\\\\Omega_n$ (the space $\\\\Omega_n(2)$ coincides with $\\\\Omega_n$). The investigation of homotopy properties of the spaces $\\\\Omega_n$ has been started by V.I. Ruban in 1985 and followed by V.A. Koshcheev, A.M. Pasko. In particular V.A. Koshcheev has proved that the spaces $\\\\Omega_n$ are simply connected. We generalized this result proving that all the spaces $\\\\Omega_n(m)$ are simply connected. In order to prove the simply connectedness of the space $\\\\Omega_n(m)$ we consider the 1-skeleton of this space. Using 1-cells we form the closed ways that create the fundamental group of the space $\\\\Omega_n(m)$. Using 2-cells we show that all these closed ways are equivalent to the trivial way. So the fundamental group of the space $\\\\Omega_n(m)$ is trivial and the space $\\\\Omega_n(m)$ is simply connected.\",\"PeriodicalId\":52827,\"journal\":{\"name\":\"Researches in Mathematics\",\"volume\":\"13 3\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-07-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Researches in Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.15421/242207\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Researches in Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.15421/242207","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
In the present paper the spaces $\Omega_n(m)$ are considered. The spaces $\Omega_n(m)$, introduced in 2018 by A.M. Pasko and Y.O. Orekhova, are the generalization of the spaces $\Omega_n$ (the space $\Omega_n(2)$ coincides with $\Omega_n$). The investigation of homotopy properties of the spaces $\Omega_n$ has been started by V.I. Ruban in 1985 and followed by V.A. Koshcheev, A.M. Pasko. In particular V.A. Koshcheev has proved that the spaces $\Omega_n$ are simply connected. We generalized this result proving that all the spaces $\Omega_n(m)$ are simply connected. In order to prove the simply connectedness of the space $\Omega_n(m)$ we consider the 1-skeleton of this space. Using 1-cells we form the closed ways that create the fundamental group of the space $\Omega_n(m)$. Using 2-cells we show that all these closed ways are equivalent to the trivial way. So the fundamental group of the space $\Omega_n(m)$ is trivial and the space $\Omega_n(m)$ is simply connected.