A theorem of James and Thomas (Note on the classification of cross-sections, Topology 4) asserts that the space of immersions, up to regular homotopy, from a compact surface $S$ into $\mathbb{R}^3$ has $2^{2-\chi(S)}$ connected components (where $\chi(S)$ is the Euler characteristic of $S$), but I could not find a reference in the case of punctured surfaces. So I wanted to know if there was anything known related to this extension.
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asked Jan 19, 2017 at 15:17
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2$\begingroup$ If you're willing to assume oriented, note that this is the same as the space of bundle monomorphisms over $\vee_k S^1$ between the trivial bundles of rank 2 and rank 3, respectively; this, then, is the same as $SO(3) \times \prod_k \Omega SO(3)$. (Here $k$ is $1-\chi$.) So there are $2^{1-\chi}$ components. $\endgroup$– mmeJan 19, 2017 at 16:30
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$\begingroup$ Thank you very much Mr. Miller. You can post what you wrote as an answer, and I will wait for some time before I accept it, to see if there is a complete known answer. $\endgroup$– Paul-BenjaminJan 19, 2017 at 16:42
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