This is the beginning of an obstruction theory for deciding whether a map is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall’s self-intersection number which tells the whole story in higher dimensions. Our second order obstruction is defined if vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of modulo –symmetry (rather then just one copy modulo –symmetry). It generalizes to the non-simply connected setting the Kervaire–Milnor invariant which corresponds to the Arf–invariant of knots in 3–space.
We also give necessary and sufficient conditions for moving three maps to a position in which they have disjoint images. Again the obstruction generalizes Wall’s intersection number which answers the same question for two spheres but is not sufficient (in dimension ) for three spheres. In the same way as intersection numbers correspond to linking numbers in dimension 3, our new invariant corresponds to the Milnor invariant , generalizing the Matsumoto triple to the non simply-connected setting.
"Higher order intersection numbers of 2–spheres in 4–manifolds." Algebr. Geom. Topol. 1 (1) 1 - 29, 2001. https://doi.org/10.2140/agt.2001.1.1