I'm interested in proving the following fact, which seems to naturally arise from gradient flow deformations, but appears to be a bit tricky.
Consider a smooth family $$f_s : M \to \mathbb{R}, \quad s \in [0,1]$$ of smooth functions on a compact manifold $M$ with boundary $\partial M$. Suppose that for any $s \in [0,1]$, $0$ is a regular value of $f_s$, and $f_s^{-1}(0)$ is transversal $\partial M$. This implies that for any $s \in [0,1]$, the spaces $f_s^{-1}(0)$ and $f_s^{-1}(0) \cap \partial M$ are manifolds. So far, so good. Now, I'm interested in the negative spaces $$f_s^- := f^{-1}(-\infty, 0]), \quad \partial f_s^- := f^{-1}(-\infty, 0]) \cap \partial M.$$ More precisely, I would like to find a homotopy equivalence of pairs $$(f_0^-, \partial f_0^-) \simeq (f_1^-, \partial f_1^-).$$ I think that, by using gradient flow deformations, I could find a homotopy equivalence $f_0^- \simeq f_1^-$. However, building such a map in a way that it preserves the boundary $\partial f_s^-$ doesn't seem so obvious to me.
Another similar exercise is the following: suppose that we only have one function $f : M \to \mathbb{R}$, and a family $N_s \subset M$ of submanifolds with boundary $\partial N_s$, such that $f^{-1}(0)$ is transversal to $N_s$ and $\partial N_s$ for all $s$. Then there exists a homotopy equivalence $$(f^-_{|N_0}, f^-_{|\partial N_0}) \simeq (f^-_{|N_1}, f^-_{|\partial N_1}).$$ Is it possible to treat both cases in the same proof ?
