TITLE: Spin 4-manifold bounded by a mapping torus of tori
QUESTION [10 upvotes]: Consider a smooth torus endowed with the non-bounding spin structure. Pick a basis of its first homology and a diffeomorphism inducing the S-transformation
$\left(\begin{array}{cc} 0 & 1 \\-1 & 0\end{array}\right)$
in the chosen basis. Construct the mapping torus $U$ of this diffeomorphism. Take the non-bounding spin structure on the $S^1$ base. This induces a spin structure on $U$. As the 3rd spin cobordism group vanishes, there is a smooth spin 4-manifold $W$ whose boundary is $U$.
The question is: How can one concretely construct such a $W$? 
More generally, when given a manifold that is known to bound, what are the tools available to construct a concrete bordism to the empty manifold?
Update: I explain why Marco's answer below does not provide 4-manifolds bounding the mapping torus above with the required spin structure. It is not difficult to see that $H^1(U;\mathbb{Z}_2) = \mathbb{Z}_2^2$, with one generator associated to the base circle and one generator associated to the cycle on the torus fiber left invariant under the S transformation. The required spin structure restricts to the non-bounding spin structure on representative loops in both classes. This means that these loops cannot become homotopically trivial in the 4-manifold. As the 4-manifolds Marco constructs are all simply connected, they cannot carry a spin structure restricting to the one I am interested in on the boundary.

REPLY [4 votes]: Marco's answer above comes close to a solution, but as he mentions, one still has to figure out which of the manifolds bounds the mapping torus with the specified spin structure. I want to record this part of the reasoning in this answer.
The possible four manifolds are described by the following diagrams:

Marco's second diagram above.
A diagram with a trivalent node labeled 0, linked to a -4 node, a -2 node and a string of three 2 nodes, obtained from the characteristic sublink given by one of the -4 nodes in Marco's first diagram.
The same diagram as above, obtained from the other -4 node.
An affine E7 Dynkin diagram, obtained from the characteristic sublink given by the three nodes -2, -4, -4 in Marco's first diagram.

The Rokhlin invariant of a spin 3-manifold is given by the signature mod 16 of a spin 4-manifold admitting it as a boundary. When the 4-manifold is specified by a plumbing diagram as above, this is just the signature of the adjacency matrix of the diagram (where the links indicate an off-diagonal value of -1). For reasons that will become clear, we only need Rokhlin invariants mod 8. The Rokhlin invariants mod 8 of the four 4-manifolds above are -1,1,1,-1.
Now we need a way of computing the Rokhlin invariant directly on the mapping torus to compare with the values above. A method for computing the Rokhlin invariant mod 8 in this way has been described by Taylor in 

Taylor, L. R, Relative Rochlin invariants, Topology and its Applications, 1984, 18, 259 - 280

The Rokhlin invariant is the Arf invariant of a quadratic refinement of the linking pairing on the torsion part of the first integral homology. The value of the quadratic refinement on a given homology class is determined by the restriction of the spin structure of the 3-manifold to a representing cycle. Getting all the sign conventions straight to compare with the values of the Rokhlin invariant computed above looks like a nightmare, but fortunately we can reason as follows.
On the one hand, the first torsion homology can be represented by one of the cycle generating the homology of the fiber of $U$. This means that switching the spin structure on the base circle will not change the Rokhlin invariant. On the other hand, Johannes' solution above corresponds to the affine E7 diagram, as it is the only one that can accomodate the degree 2 cohomology classes that will necessarily appear when resolving the orbifold singularities. As I mentioned in a comment of his solution, he has the right spin structure on the fibers, but not on the base. This means that the 4-manifold that I am after is the one that has the same Rokhlin invariant, namely the one originally proposed by Marco.