Immersion extension-lift over a Morse function (Q627710)

Let \(V\) be a compact connected oriented surface with boundary, \(f:\partial V\times [0,1) \to\mathbb R\) a Morse function and \(\iota: \partial V\times [0,1) \to V\) a collaring. An immersion extension-lift over \(f\) is an orientation preserving immersion \(F:V\to\mathbb R^2\) such that \(f= \pi\circ F\circ \iota\), where \(\pi:\mathbb R^2\to\mathbb R\) is the orthogonal projection. The paper gives necessary and sufficient conditions for the existence of an immersion extension-lift over a given \(f.\) These conditions consist of three statements which are too technical to be stated here. Furthermore, let \(\text{Imm}_f(V,\mathbb R^2)\) denote the set of immersion extension-lifts over \(f\). Then \(F,F'\in\text{Imm}_f(V,\mathbb R^2)\) are \(f\)-image homotopic if there exist a family of immersion extension-lifts \(F_t\in\text{Imm}_f(V,\mathbb R^2)\), \(t\in[0,1]\) and an orientation preserving diffeomorphism \(h:V\to V\) such that \(h|\partial V\times [0,1) = \text{id}\), \(F_0=F\) and \(F_1=F'\circ h\). A criterion is given when two immersion extension-lifts \(F\) and \(F'\) over \(f\) are \(f\)-image homotopic. The third result studies relations between image homotopy classes when \(f\) is deformed under a Morse homotopy.