1. Introduction -- 2. Basic lemmas -- 3. A look ahead -- 4. Toral chains of open 3-cells -- 5. Partitions of 2-spheres -- 6. Notation -- 7. Preliminary lemmas -- 8. Adjusting 2-spheres -- 9. The Flange Lemma -- 10. Constructing punctured discs -- 11. The 2-sphere construction lemma -- 12. The adjustment lemma -- 13. A special case of the main lemma -- 14. The main lemma -- 15. The main result -- 16. Construction of K -- 17. Construction of K' -- 18. Applications of Theorem 1 -- 19. Linking in decomposition spaces -- References