Recursive coboundary formula for cycles in acyclic chain complexes

Tomasz Kaczyński

DOI: http://dx.doi.org/10.12775/TMNA.2001.039

Abstract


Given an $(m-1)$-dimensional cycle $z$ in a finitely generated acyclic chain complex,
we want to explicitly construct an $m$-dimensional chain $\cob(z)$ 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.

Keywords


Homology computation; cycle; coboundary; algorithm

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