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# The Topological Interpretation of the Core Group of a Surface in $S^4$

Published:2002-03-01
Printed: Mar 2002
• Józef H. Przytycki
• Witold Rosicki
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## Abstract

We give a topological interpretation of the core group invariant of a surface embedded in $S^4$ \cite{F-R}, \cite{Ro}. 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.
 MSC Classifications: 57Q45 - Knots and links (in high dimensions) {For the low-dimensional case, see 57M25} 57M12 - Special coverings, e.g. branched 57M05 - Fundamental group, presentations, free differential calculus