This is slightly related to my previous question Higher homotopy groups of a string complement in a cylinder., but now I'm only interested in a following one:
We call the image of a smooth embedding $f:\coprod_{i=1}^n [0,1]\times\{i\} \hookrightarrow \mathbb{D}^2 \times [0,1] $ a string if $f(0,i)$ and $f(1,i)$ have the second coordinates $0$ and $1$ respectively, and also, $f$ sends interior to the interior. Let us denote the complement of a tubular neighborhood of a string by $X_f$. Let $p:X_f\rightarrow [0,1]$ be a projection of the first coordinate. Then there is an inclusion $i:B\hookrightarrow X_f$, where $B:=p^{-1}(0)$ is a kind of a left boundary of a cylinder.
How to show that $i$ induces isomorphism on first homology groups and a surjection on the second homology groups? (I don't know if it is true, but it feels so.)
I looks like relative homology sequences and Mayer-Vietoris sequences are not helpful. Maybe some bordism arguments need to be used?