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

DOI: 10.2140/pjm.1973.49.127

Abstract

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 (n− 2)-cycle on L. For each nonnegative integer d, the d-th elementary ideal E_d(l) is associated to l on L in Sⁿ. If l and l′ are homologous on L, then E_d(l) is equal to E_d(l′). Now the collection of E_d(l) for all possible l is a topological invariant of L in Sⁿ.

In this paper the following two cases of E_d(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

Milestones

Received: 3 August 1972

Published: 1 November 1973