{"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}
引用次数: 8
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
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