Study of factorization of stable map using lift, extension and extension-lift

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For an oriented once punctured surface, we consider a Morse function of collar neighborhood of the boundary to the line. We study the question when this Morse function has an extension-lift of the original surface to the plane. We give a judgment of existent and non-existent law.We consider the relationship between an immersion of an oriented surface into the 3-space and a stable map into the plane. When we fix a stable map into the plane, we classify immersions which are lifts of this stable map up to regular homotopy.