论文标题
平滑近似及其应用于同质类型
Smooth approximations and their applications to homotopy types
论文作者
论文摘要
令$ m，n $平滑的多种流形，$ \ MATHCAL {c}^{r}（m，n）$ $ {c}^{r} $映射的空间，这些空间与弱$ c^{r} $ c^{r} $ whitney topology和$ \ supcal {b} \ subset {b} \ subset \ mathcal untote事实证明，$ 0 \ leq r <s \ leq \ infty $包含$ \ mathcal {b} \ cap \ mathcal {c}^{s}^{s}（m，n）\ subset \ subset \ mathcal {b} $是一个较弱的同型等值。还建立了这种结果的参数化变体。特别是，这表明，对于紧凑的歧管$ m $ \ Mathrm {id} _ {M}）$ \ m artercal {c}^{c}^{r} $ diffeomorphisms $ m $ at $ \ mathrm {id} _ {m} $是一个弱的同质副本。
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 $Ω(\mathcal{D}^{r}(M), \mathrm{id}_{M})$ of the group of $\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\mathrm{id}_{M}$ is a weak homotopy equivalence.
