论文标题
稳定映射密度的旧结果和新结果
Old and new results on density of stable mappings
论文作者
论文摘要
稳定地图的密度是本文的共同线程。我们回顾惠特尼对$ c^{\ infty} $和$ c^{0} $稳定性的可区分映射和thomather理论的奇异性的贡献。提出了无限的和代数方法，以证明定理A和定理B在适当的稳定和拓扑稳定的映射的密度上，$ f：n^{n} \ to p^{p}。$定理，$ a $ n的$ n $ $ p）是$ $ p），如果是$ p），如果是$ p）。 \ emph {nice dimensions，}，而定理B表示拓扑稳定的地图的密度可容纳任何一对$（N，p）。 Thommather地图是一张拓扑稳定的地图$ f：n \至P $，其与之相关的$ k $ -jet映射$ j^{k} f：n \ to p $ to to $ j^{k^{k}（n，p）中的汤姆·梅瑟分层。映射是为了确定对Lipschitz稳定映射密集的$（n，p）$。我们讨论了Nguyen，Ruas和Trivedi对这一主题的最新结果，这是对Lipschitz稳定映射密度在良好维度边界中的密度进行的猜想。在最后一部分中，Damon的结果与$ \ MATHCAL {A} $ - MAP-GERMS和$ \ MATHCAL {K} _ {V} $分类进行了审查和开放问题的稳定外向问题。
Density of stable maps is the common thread of this paper. We review Whitney's contribution to singularities of differentiable mappings and Thom-Mather theories on $C^{\infty}$ and $C^{0}$-stability. Infinitesimal and algebraic methods are presented in order to prove Theorem A and Theorem B on density of proper stable and topologically stable mappings $f:N^{n}\to P^{p}.$ Theorem A states that the set of proper stable maps is dense in the set of all proper maps from $N$ to $P$, if and only if the pair $(n,p)$ is in \emph{nice dimensions,} while Theorem B shows that density of topologically stable maps holds for any pair $(n,p).$ A short review of results by du Plessis and Wall on the range in which proper smooth mappings are $C^{1}$- stable is given. A Thom-Mather map is a topologically stable map $f:N \to P$ whose associated $k$-jet map $j^{k}f:N \to P$ is transverse to the Thom-Mather stratification in $J^{k}(N,P).$ We give a detailed description of Thom-Mather maps for pairs $(n,p)$ in the boundary of the nice dimensions.The main open question on density of stable mappings is to determine the pairs $(n,p)$ for which Lipschitz stable mappings are dense. We discuss recent results by Nguyen, Ruas and Trivedi on this subject, formulating conjectures for the density of Lipschitz stable mappings in the boundary of the nice dimensions. At the final section, Damon's results relating $\mathcal{A}$-classification of map-germs and $\mathcal{K}_{V}$ classification of sections of the discriminant $V=Δ(F)$ of a stable unfolding of $f$ are reviewed and open problems are discussed.
