{"title":"光滑逼近及其在同伦类型上的应用","authors":"Олександра Олександрівна Хохлюк, S. Maksymenko","doi":"10.15673/tmgc.v13i2.1781","DOIUrl":null,"url":null,"abstract":"Let $M, N$ the be smooth manifolds, $\\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with weak $C^{r}$ Whitney topology, and $\\mathcal{B} \\subset \\mathcal{C}^{r}(M,N)$ an open subset. It is proved that for $0\\leq r<s\\leq\\infty$ the inclusion $\\mathcal{B} \\cap \\mathcal{C}^{s}(M,N) \\subset \\mathcal{B}$ is a weak homotopy equivalence. It is also established a parametrized variant of such a result. In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $\\mathcal{C}^{s}$ isotopies $[0,1]\\times M \\to M$ fixed near $\\{0,1\\}\\times M$ into the space of loops $\\Omega(\\mathcal{D}^{r}(M), \\mathrm{id}_{M})$ of the group of $\\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\\mathrm{id}_{M}$ is a weak homotopy equivalence.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"112 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":"{\"title\":\"Smooth approximations and their applications to homotopy types\",\"authors\":\"Олександра Олександрівна Хохлюк, S. Maksymenko\",\"doi\":\"10.15673/tmgc.v13i2.1781\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $M, N$ the be smooth manifolds, $\\\\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with weak $C^{r}$ Whitney topology, and $\\\\mathcal{B} \\\\subset \\\\mathcal{C}^{r}(M,N)$ an open subset. It is proved that for $0\\\\leq r<s\\\\leq\\\\infty$ the inclusion $\\\\mathcal{B} \\\\cap \\\\mathcal{C}^{s}(M,N) \\\\subset \\\\mathcal{B}$ is a weak homotopy equivalence. It is also established a parametrized variant of such a result. In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $\\\\mathcal{C}^{s}$ isotopies $[0,1]\\\\times M \\\\to M$ fixed near $\\\\{0,1\\\\}\\\\times M$ into the space of loops $\\\\Omega(\\\\mathcal{D}^{r}(M), \\\\mathrm{id}_{M})$ of the group of $\\\\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\\\\mathrm{id}_{M}$ is a weak homotopy equivalence.\",\"PeriodicalId\":8433,\"journal\":{\"name\":\"arXiv: Algebraic Topology\",\"volume\":\"112 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-08-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"8\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Algebraic Topology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.15673/tmgc.v13i2.1781\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.15673/tmgc.v13i2.1781","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Smooth approximations and their applications to homotopy types
Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with weak $C^{r}$ Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset. It is proved that for $0\leq r
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