论文标题
将地图近似为真实代数同质空间
On approximation of maps into real algebraic homogeneous spaces
论文作者
论文摘要
令X为紧凑的（分别紧凑且非词性）真实代数品种，让Y成为某些线性真实代数群的均匀空间。我们证明，当且仅当它与常规映射是同型时，只有在c^o（resp。C^Infinity）拓扑中的常规地图可以通过常规地图近似f：x-> y的连续（c^Infinity）映射。乘以y = s^p，是单位p维球体，我们获得了自1980年代以来已经打开的几个问题的解决方案，并且涉及单位球中的地图的近似值。这对于单位球之间的地图近似产生了几个后果。例如，我们证明，对于每个正整数n，每个c^infinity映射从s^n到s^n都可以通过c^Infinity拓扑中的常规图近似。到目前为止，这种结果仅以五个特殊值N，即n = 1,2,3,4或7。
Let X be a compact (resp. compact and nonsingular) real algebraic variety and let Y be a homogeneous space for some linear real algebraic group. We prove that a continuous (resp. C^infinity) map f:X-->Y can be approximated by regular maps in the C^o (resp. C^infinity) topology if and only if it is homotopic to a regular map. Taking Y=S^p, the unit p-dimensional sphere, we obtain solutions of several problems that have been open since the 1980's and which concern approximation of maps with values in the unit spheres. This has several consequences for approximation of maps between unit spheres. For example, we prove that for every positive integer n every C^infinity map from S^n into S^n can be approximated by regular maps in the C^infinity topology. Up to now such a result has only been known for five special values of n, namely, n=1,2,3,4 or 7.
