Odd-gon relations and their cohomology

A cohomology theory for "odd polygon" relations -- algebraic imitations of Pachner moves in dimensions 3, 5, ... -- is constructed. Manifold invariants based on polygon relations and nontrivial polygon cocycles are proposed. Example calculation results are presented.
Comment: 32 pages. v2: proof of Proposition 13 made more accurate; new calculation results added to Table 2. This article supersedes arXiv:2110.08780