This proposition in Bott & Tu have been haunting me for a year or so since I always have to come back to this book for references. More precisely, the second equality in Proposition 12.8 in page 135 of Differential Forms in Algebraic Topology has a weird assertion.
Let $E \twoheadrightarrow X$ be a vector bundle over a manifold $X$, $S_0$ the image of the zero section, $S$ the image of a section transversal to $S_0$, $Z = S \cap S_0$, $x \in Z$ (by identifying $S_0$ and $X$) and $S_x = (N_{Z/S})_x$.
Let $\Phi$ be the Thom class of $N_{Z/X}$. The authors claim that $$\int_{S_x} \Phi = \int_{E_x} \Phi$$, because $S_x$ and $E_x$ are homotopic modulo the region in $E$ where $\Phi$ is zero.
In this context, this justification makes no sense. The unique possible theorem that they're alluding to is contained in the answer of invariance of integrals for homotopy equivalent spaces . However this still makes no sense in the equality above even by fixing a homotopy equivalence $f: S_x \rightarrow E_x$ since $f^* \Phi \neq \Phi$ might happens.
I would like a clarification of the equality mentioned above.
Thanks in advance.