1. Introduction Part 1. Controlled connectivity and openness results 2. Outline, main results and examples 3. Technicalities concerning the $CĈ{n-1}$ property 4. Finitary maps and sheaves of maps 5. Sheaves and finitary maps over a control space 6. Construction of sheaves with positive shift 7. Controlled connectivity as an open condition 8. Completion of the proofs of Theorems A and A 9. The invariance theorem Part 2. The geometric invariants 10. Outline, main results and examples 11. Further technicalities on CAT(0) spaces 12. $CĈ{n-1}$ over endpoints 13. Finitary contractions towards endpoints 14. From $CĈ{n-1}$ over endpoints to contractions 15. Proofs of Theorems E-H
