平面连通域到CxC * 的映射同伦类的真全纯浸入
Proper holomorphic immersions in homotopy classes of maps from finitely connected planar domains into CxC*
摘要：
格罗莫夫在他1989年关于奥卡原理的开创性论文中证明了从Stein流形到椭圆流形的每一个连续映射都同伦于全纯映射。以前，我们已经表明，给定一个连续的地图$X \to \C\times\C^*$从一个连续连通的平面区域$X$没有孤立的边界点，一个更强的Oka性质，即该地图是同伦的一个适当的全纯嵌入。在这里，我们表明，每个连续映射从一个连通的平面域，可能与穿刺，到$\C\times\C^*$是同伦的适当的浸入，确定在最多可数许多对不同的点，在大多数情况下，只有1000对。通过研究浸入是内射的情形，我们得到了几类具有孤立边界点的平面域的嵌入的一个强Oka性质.目前尚不清楚是否有一个强的Oka属性嵌入持有一般域时，孤立的边界点。最后，我们给出了一些关于零同伦真全纯嵌入$\C^* \to \C\times\C^*$存在性的观察。
英文摘要：
Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. Previously we have shown that, given a continuous map $X \to \C\times\C^*$ from a finitely connected planar domain $X$ without isolated boundary points, a stronger Oka property holds, namely that the map is homotopic to a proper holomorphic embedding. Here we show that every continuous map from a finitely connected planar domain, possibly with punctures, into $\C\times\C^*$ is homotopic to a proper immersion that identifies at most countably many pairs of distinct points, and in most cases, only finitely many pairs. By examining situations in which the immersion is injective, we obtain a strong Oka property for embeddings of some classes of planar domains with isolated boundary points. It is not yet clear whether a strong Oka property for embeddings holds in general when the domain has isolated boundary points. We conclude with some observations on the existence of a null-homotopic proper holomorphic embedding $\C^* \to \C\times\C^*$.
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