  Bott and Taubes constructed knot invariants by integrating differential forms
along the fiber of a bundle over the space of knots, generalizing the Gauss
linking integral. Their techniques were later used to construct real cohomology
classes in spaces of knots and links in higher-dimensional Euclidean spaces. In
previous work, we constructed cohomology classes in knot spaces with arbitrary
coefficients by integrating via a Pontrjagin--Thom construction. We carry out a
similar construction over the space of string links, but with a refinement in
which configuration spaces are glued together according to the combinatorics of
weight systems. This gluing is somewhat similar to work of Kuperberg and
Thurston. We use a formula of Mellor for weight systems of Milnor invariants,
and we thus recover the Milnor triple linking number for string links, which is
in some sense the simplest interesting example of a class obtained by this
gluing refinement of our previous methods. Along the way, we find a description
of this triple linking number as a "degree" of a map from the 6-sphere to a
quotient of the product of three 2-spheres.
