Recursive coboundary formula for cycles in acyclic chain complexes

Abstract

Given an $(m-1)$-dimensional cycle $z$ in a finitely generated acyclic chain complex, we want to explicitly construct an $m$-dimensional chain ${\rm Cob}(z)$ whose algebraic boundary is $z$. The acyclicity of the chain complex implies that a solution exists (it is not unique) but the traditional linear algebra methods of finding it lead to a high complexity of computation. We are searching for more efficient algorithms based on geometric considerations. The main motivation for studying this problem comes from the topological and computational dynamics, namely, from designing general algorithms computing the homomorphism induced in homology by a continuous map. This, for turn, is an essential step in computing such invariants of dynamical properties of nonlinear systems as Conley index or Lefschetz number. Another potential motivation is in the relationship of our problem to the problem of finding minimal surfaces of closed curves.