Vol. 103, No. 2, 1982

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Knot groups in S4 with nontrivial homology

Andrew Michael Brunner, Edward James Mayland, Jr. and Jonathan Simon

Vol. 103 (1982), No. 2, 315–324

In this paper we exhibit smooth 2-manifolds F2 in the 4-sphere S4 having the property that the second homology of the group π1(S4 F2) is nontrivial. In particular, we obtain tori for which H2(π1)Z2 and, by forming connected sums, surfaces of genus n for which H2(π1) is the direct sum of n copies of Z2. Corollaries include: (1) There are knotted surfaces in S4 that cannot be constructed by forming connected sums of unknotted surfaces and knotted 2-spheres. (2) The class of groups that occur as knot groups of surfaces in S4 is not contained in the class of high dimensional knot groups of Sn in Sn+2.

Mathematical Subject Classification 2000
Primary: 57Q45
Received: 4 June 1978
Revised: 19 November 1980
Published: 1 December 1982
Andrew Michael Brunner
Edward James Mayland, Jr.
Jonathan Simon