Vol. 49, No. 1, 1973

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On elementary ideals of 𝜃-curves in the 3-sphere and 2-links in the 4-sphere

Shin’ichi Kinoshita

Vol. 49 (1973), No. 1, 127–134

Let L be a polyhedron in an n-sphere lSn(n S) that does not separate Sn. A topological invariant of the position of L in Sn can be introduced as follows: Let l be an integral (n2)-cycle on L. For each nonnegative integer d, the d-th elementary ideal Ed(l) is associated to l on L in Sn. If l and lare homologous on L, then Ed(l) is equal to Ed(l). Now the collection of Ed(l) for all possible l is a topological invariant of L in Sn.

In this paper the following two cases of Ed(l) are considered: (1) l is a l-cycle on a 𝜃-curve L in S3, and (2) l is a 2-cycle on a 2-link L in S4, i.e., the union of two disjoint 2-spheres in S4, where each of two 2-spheres is trivially imbedded in S4.

Mathematical Subject Classification
Primary: 55A25
Received: 3 August 1972
Published: 1 November 1973
Shin’ichi Kinoshita