{"title":"乱序单形的拓扑结构","authors":"Dmitry N. Kozlov","doi":"10.1007/s40062-018-0214-6","DOIUrl":null,"url":null,"abstract":"<p>In this paper we define a?family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a?standard <i>d</i>-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a?<span>\\(\\Delta \\)</span>-complex. As our main result, we completely determine the homotopy type of these spaces. In fact, somewhat surprisingly, we are able to prove that each of them is either contractible or homotopy equivalent to an?odd-dimensional sphere. We develop the language to determine the homotopy type directly from the combinatorics of the indexing word. As added benefit of our investigation, we are able to emulate the Dunce hat phenomenon, and to obtain a?large family of both <span>\\(\\Delta \\)</span>-complexes, as well as simplicial complexes, which are contractible, but not collapsible.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"371 - 391"},"PeriodicalIF":0.7000,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0214-6","citationCount":"3","resultStr":"{\"title\":\"Topology of scrambled simplices\",\"authors\":\"Dmitry N. Kozlov\",\"doi\":\"10.1007/s40062-018-0214-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper we define a?family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a?standard <i>d</i>-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a?<span>\\\\(\\\\Delta \\\\)</span>-complex. As our main result, we completely determine the homotopy type of these spaces. In fact, somewhat surprisingly, we are able to prove that each of them is either contractible or homotopy equivalent to an?odd-dimensional sphere. We develop the language to determine the homotopy type directly from the combinatorics of the indexing word. As added benefit of our investigation, we are able to emulate the Dunce hat phenomenon, and to obtain a?large family of both <span>\\\\(\\\\Delta \\\\)</span>-complexes, as well as simplicial complexes, which are contractible, but not collapsible.</p>\",\"PeriodicalId\":49034,\"journal\":{\"name\":\"Journal of Homotopy and Related Structures\",\"volume\":\"14 2\",\"pages\":\"371 - 391\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2018-09-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s40062-018-0214-6\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Homotopy and Related Structures\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40062-018-0214-6\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Homotopy and Related Structures","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40062-018-0214-6","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
In this paper we define a?family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a?standard d-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a?\(\Delta \)-complex. As our main result, we completely determine the homotopy type of these spaces. In fact, somewhat surprisingly, we are able to prove that each of them is either contractible or homotopy equivalent to an?odd-dimensional sphere. We develop the language to determine the homotopy type directly from the combinatorics of the indexing word. As added benefit of our investigation, we are able to emulate the Dunce hat phenomenon, and to obtain a?large family of both \(\Delta \)-complexes, as well as simplicial complexes, which are contractible, but not collapsible.
期刊介绍:
Journal of Homotopy and Related Structures (JHRS) is a fully refereed international journal dealing with homotopy and related structures of mathematical and physical sciences.
Journal of Homotopy and Related Structures is intended to publish papers on
Homotopy in the broad sense and its related areas like Homological and homotopical algebra, K-theory, topology of manifolds, geometric and categorical structures, homology theories, topological groups and algebras, stable homotopy theory, group actions, algebraic varieties, category theory, cobordism theory, controlled topology, noncommutative geometry, motivic cohomology, differential topology, algebraic geometry.