  This paper is motivated by recent developments in group stability, high
dimensional expansion, local testability of error correcting codes and
topological property testing. In Part I, we formulate and motivate three
stability problems: 1. Homomorphism stability: Are almost homomorphisms close
to homomorphisms? 2. Covering stability: Are almost coverings of a cell complex
close to genuine coverings of it? 3. Cocycle stability: Are 1-cochains whose
coboundary is small close to 1-cocycles? We then prove that these three
problems are equivalent.
