The topological interpretation of the core group of a surface in S^4

Abstract

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