Acyclicity of a pair in Morse Theory. Let $\delta\colon M\times\mathbb{R}^{2N}\to\mathbb{R}$ be a smooth function such that outside a compact set, one has:
$$\delta(x,e_1,e_2)=A(e_1)-A(e_2),$$
where $A\colon\mathbb{R}^N\to\mathbb{R}$ is a nonzero linear form. The map $\delta$ also satisfies $\delta(x,e_1,e_2)=-\delta(x,e_2,e_1)$, so that its critical points come into pair and its critical value set is symmetric.
Let $\omega>\varepsilon>0$ such that all positive critial values of $\delta$ are stricly between $\varepsilon$ and $\omega$. 
For $a\in\mathbb{R}$, let $\delta^a:=\delta^{-1}(]-\infty,a])$. It is claimed in a paper (proof of Theorem 6.1(1), page 2472) that:

The pair $(\delta^\omega,\delta^{-\omega})$ is acyclic, its homology vanishes.

Not only I did not manage to see why this is true, but I find this claim weird. Indeed, $\delta^{-\omega}$ is acyclic and from the relative homology long exact sequence, one has $H_\bullet(\delta^\omega,\delta^{-\omega})\cong H_\bullet(\delta^\omega)$, so that $\delta^\omega$ will also be acyclic, but it is a deformation retract of $M\times\mathbb{R}^{2N}$ which needs not to be acylic. Am I going mad?
Any enlightenment will be greatly appreciated!
 A: Let us first introduce the map $B\colon\mathbb{R}^N\times\mathbb{R}^N\to\mathbb{R}$ defined by:
$$B(e_1,e_2)=A(e_1)-A(e_2).$$
Using the behaviour at the infinity of $\delta$, there exists $\delta^c\colon X\times\mathbb{R}^{2N}\to\mathbb{R}$ with compact support such that:
$$\delta(x,e_1,e_2)=\delta^c(x,e_1,e_2)+B(e_1,e_2).$$
Then, choosing a bigger $\omega$ if needed, the levels $\pm\omega$ stay regular along the homotopy $s\mapsto s\delta^c+B$, therefore using a parametric version of the standard critical non-crossing result in Morse theory, the sublevels $\delta^{\pm\omega}$ and $B^{\pm\omega}$ are deformation retracts, so that:
$$H(\delta^\omega,\delta^{-\omega})=H(B^\omega,B^{-\omega})$$
Besides, since $B$ has no critical point, the standard critical non-crossing result in Morse theory shows that $B^\omega$ and $B^{-\omega}$ are deformation retracts and $(B^\omega,B^{-\omega})$ is acyclic, whence the result.

This is the lemma 3.12 in the article Obstruction to Lagrangian cobordisms between Legendrians via generating families by Joshua Sabloff and Lisa Traynor.
