  We compute the abelianisations of the mapping class groups of the manifolds
$W_g^{2n} = g(S^n \times S^n)$ for $n \geq 3$ and $g \geq 5$. The answer is a
direct sum of two parts. The first part arises from the action of the mapping
class group on the middle homology, and takes values in the abelianisation of
the automorphism group of the middle homology. The second part arises from
bordism classes of mapping cylinders and takes values in the quotient of the
stable homotopy groups of spheres by a certain subgroup which in many cases
agrees with the image of the stable $J$-homomorphism. We relate its calculation
to a purely homotopy theoretic problem.
