Consider a manifold $M$ of dimension $4k + 2$, $k$ an integer. Pick a diffeomorphism $\phi$ of $M$ and construct the mapping torus $T$ of $\phi$. Suppose that there is a $4k+4$ dimensional manifold $B$ admitting $T$ as a boundary.

My question is: Has the signature of $B$ been computed somewhere, at least in some class of examples? I would be happy with the signature modulo an integer that is a multiple of 4.

I would suspect that the signature modulo 4 should depend only on the class of $\phi$ in the mapping class group of $M$ and that the computation involved is purely algebraic...

Thanks in advance.