Immersions of surfaces with boundary into the plane. (Q1880114)

Let \(M\) be a compact connected oriented surface with boundary. \textit{L. Kaufman} characterized all orientation preserving immersions of \(M\) into the plane up to image homotopy in [Illinois J. Math. 23, 684--665 (1978; Zbl 0425.57007)], where two orientation preserving immersions \(f\) and \(g\) of \(M\) into the plane are said to be image homotopic if there exists a diffeomorphism \(f:M\to M\) such that \(f\) is regularly homotopic to \(gh\). But the result is incorrect in the case when \(M\) is a torus with \(k (k>1)\) boundary components. The author of this paper corrects this result and gives a complete invariant of image homotopy classes in the above case.