I am trying to do an exercise that is the following :
Suppose that $A\cup B$ is a compact $n-$dimensional manifold such that $A,B\subset A\cup B $ are compact $n$-dimensional submanifolds and $A\cap B$ is a compact $n-1$ dimensional submanifold. Show that $$\chi(A\cup B)=\chi(A)+\chi(B)-\chi(A\cap B)$$
Now first since there are many definitions for $\chi(A\cup B)$, I will say with which one I am working with . Let $f:M\rightarrow TM$ be a section transverse to the zero section. Then for each $x\in M$ we have a canonical identification $T_{(x,0)}(TM)\cong T_x M$ and so $Tf$ induces an automorphism $\phi_x$ from $T_xM$ to $T_x M$ and we define $Ind_x f=\frac{Det \phi_x}{|Det \phi_x|}$.
The way we compute $\chi(A\cup B)$ is just to take a transversal section to the zero section and compute the $Ind_x f$, and then we can use this $f$ also to compute $\chi(A)$ and $\chi(B)$. Now for $\chi(A\cap B)$ I am not interely sure what section to use that is related to $f$.
Any help is aprecciated.
(I didn't include the definition for manifolds with boundary because I am trying to treat this simpler case fist without having to worry about outward pointing vector fields).
As written, your question is ill-posed since you did not define what you mean by index for manifolds with boundary. One such definition can be found in the paper
Benoît Jubin, A generalized Poincaré-Hopf index theorem, arXiv:0903.0697, 2009.
With the definition (and basic results) given there, the proof of the equality $$ \chi(A\cup B)= \chi(A)+ \chi(B) - \chi(A\cap B) $$ is straightforward, you use the same arguments as in the proof of Theorem 12 in that paper. (Isotope the vector fields on $A, B$ to make them equal on the common boundary and tangent to the common boundary and then count zeroes with sign.)