# The signature of a mapping torus

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.

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If you make a connected sum of $B$ with $\pm\mathbb{CP}^{2k+2}$ you vary the signature by $\pm 1$, so it cannot depend of $\phi$ only. Maybe $B$ has an even intersection form? –  Bruno Martelli Apr 30 '11 at 6:31
Thanks for the counterexample. Supposedly I would like some extra structure on $M$ and $B$, with appropriate compatibility conditions, that would rule out such examples. This is not yet completely clear to me. I'm just asking for litterature where the signature of mapping tori is computed. I also forgott to mention the one paper I already know, which treats the case of dimension $8k+2$ spin manifolds: projecteuclid.org/euclid.bams/1183554174 There the Rohlin invariant of the mapping torus depends only on the class of the diffeomorphism. –  Samuel Monnier May 1 '11 at 9:12