Stability of Homomorphisms, Coverings and Cocycles I: Equivalence

Abstract 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: • Homomorphism stability: Are almost homomorphisms close to homomorphisms? • Covering stability: Are almost coverings of a cell complex close to genuine coverings of it? • Cocycle stability: Are 1-cochains whose coboundary is small close to 1-cocycles? We then prove that these three problems are equivalent. In Part II of this paper CL23, we present examples of stable (and unstable) complexes, discuss various applications of our new perspective, and provide a plethora of open problems and further research directions. In another companion paper CP23, we study the stability rates of random simplicial complexes in the Linial–Meshulam model.