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.
Keywords
intersection number, 4–manifold, Whitney disk, immersed
2–sphere, cubic form
