Let $(A,\partial)$ be a surface with boundary. Pick a relative bordism $W$ from $A$ to itself, i.e. $\partial W \cong (A\cup_{\partial}-A)$. Is it true that inclusion $i:A\hookrightarrow W$ induces isomorphism on homology groups ?
I am reading an article about the special case of my question where author says: "Mayer-Vietoris sequence argument shows this relative bordism is a homology bordism". I do not quite get it. I know that bordism groups $\Omega_k(-)$ define an extraordinary homology theory so Mayer-Vietoris sequence holds. But I do not see how it can be used.
P.S. Assume everything is path-connected.
Replying to the comments below: The original question comes from the example of $X_f$ here Higher homotopy groups of a string complement in a cylinder.. So I would also accept an answer which assumes that the genus of $A$ is $0$ or more simply the bordism is given by $X_f$.