Homotopy between sections

Question

Let $f:X \to S$ be a proper $C^\infty$-morphism between real manifolds. Assume that each fiber of $f$ is connected of real dimension 2 (the fiber may not be smooth, but it is the union of smooth manifolds). Suppose that $f$ admits 2 sections $s_1, s_2: S \to X$.

Question: is there a homotopy $H: S \times [0,1] \to X$ such that $H(-,0)=s_1$ and $H(-,1)=s_2$?

Remark: I would like to look at the positive side of such a problem if it is not irretrievable wrong. In fact, the $f$ in my case is a morphism between smooth complex varieties.

Answer 1

Not in general.

Suppose $f: S^1\times T \to S^1$ is the projection, where $f$ is the first factor projection and $T = S^1 \times S^1$ is the torus. Then a section amounts to a map $S^1 \to T$ and the homotopy class of a section amounts to an element of $[S^1,T] \cong \Bbb Z \times\Bbb Z$.

So you have an easy counterexample given by letting $s_i: S^1 \to T$ be the inclusion into each factor. (Smoothness is irrelevant here.)