  We give a topological interpretation of the core group invariant of a surface
embedded in S^4. We show that the group is isomorphic to the free product of
the fundamental group of the double branch cover of S^4 with the surface as a
branched set, and the infinite cyclic group. We present a generalization for
unoriented surfaces, for other cyclic branched covers, and other codimension
two embeddings of manifolds in spheres.
  The method of computing the fundamental group of n-fold cyclic branched
covers is related to the one described in R.H.Crowell, The derived group of a
permutation representation, Adv. in Math. 53(1), 1984, 99--124.
  We use these computations in recent papers:
http://front.math.ucdavis.edu/math.GT/0302098
http://front.math.ucdavis.edu/math.GT/0309140
