Immersion extension-lift over a Morse function

Let V be a compact connected oriented surface with boundary and f : ∂ V × [ 0 , 1 ) → R a non-singular function such that f | ∂ V × { 0 } is a Morse function. Let ι : ∂ V × [ 0 , 1 ) → V be a collaring of ∂V and π : R 2 → R an orthogonal projection. In this paper, we study existence of an orientation preserving immersion F : V → R 2 such that π ∘ F ∘ ι = f . We also study image homotopy classes of F when we fix f and study relation between two image homotopy classes when f is deformed under a Morse homotopy.