Proper holomorphic immersions in homotopy classes of maps from finitely connected planar domains into CxC* Finnur LarussonTyson Ritter 32H0232E1032H3532M1732Q2832Q40holomorphic immersionholomorphic embeddingproper mapRiemann surfaceOka principleStein manifoldcircular domainplanar domainhomotopy class 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 $X\to\mathbb{C}\times\mathbb{C}^{*}$ 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 $\mathbb{C}\times\mathbb{C}^{*}$ 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 $\mathbb{C}^{*}\to\mathbb{C}\times\mathbb{C}^{*}$. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5206 10.1512/iumj.2014.63.5206 en Indiana Univ. Math. J. 63 (2014) 367 - 383 state-of-the-art mathematics http://iumj.org/access/