Graph Approach to the Computation of the Homology of Continuous Maps

Abstract

We introduce an efficient algorithm to compute the homomorphism induced in (relative) homology by a continous map. The algorithm is based on a cubical approximation of the map and the theory of multivalued maps. A software implementation of the algorithms introduced in this paper is available at [27].

Recent development in rigorous computational methods in dynamical systems

“…As in the case of flows, the remark on computing the homological (instead of cohomological) Conley index is also valid here. Note that the algorithms in [8,29,36] are capable of computing the homomorphism induced in homology by a continuous map between cubical sets.…”

“…If F satisfies some additional assumptions then F can be used to compute the homomorphism induced in homology by the map f on the index pair (see [29,36]), and thus to compute the homological Conley index with respect to f . In [33,35], an algorithm is developed for the construction of a combinatorial index pair (P 1 , P 2 ) such that P 1 \ P 2 contains an apriori given set Q ⊂ H. Given an approximate rough covering Q ⊂ H of a hypothetic periodic trajectory observed in numerical simulations, one can use the algorithm for the construction of a combinatorial index pair containing Q mentioned in Section 3.5 for a representation F of the map ϕ t computed with the use of the software [7] for some apriori chosen t > 0.…”

“…In [33,35], an algorithm is developed for the construction of a combinatorial index pair (P 1 , P 2 ) such that P 1 \ P 2 contains an apriori given set Q ⊂ H. Given an approximate rough covering Q ⊂ H of a hypothetic periodic trajectory observed in numerical simulations, one can use the algorithm for the construction of a combinatorial index pair containing Q mentioned in Section 3.5 for a representation F of the map ϕ t computed with the use of the software [7] for some apriori chosen t > 0. If the algorithm succeeds then it constructs a combinatorial index pair (P 1 , P 2 ) such that N := |P 1 \ P 2 | is an isolating neighborhood with respect to ϕ t , and the homological Conley index of N with respect to ϕ t can be automatically computed with algorithms available in [8,29,36]. By Lemma 3.5, N is an isolating neighborhood also for the flow ϕ, and S := inv(N, ϕ t ) = inv(N, ϕ).…”

“…The complexity of these algorithms is linear (i.e., they are very fast). The combinatorial Morse sets M i can be used directly for the computation of the Conley index using the approach introduced in [29] and refined in [36], where each M i gives rise to the pair M i ∪F(M i ), F(M i )\M i satisfying the conditions for a combinatorial index pair (Definition 3.6) with respect to F L . In this way, a combinatorial Conley-Morse decomposition can be effectively computed.…”

“…Throughout the paper, it is emphasized that the use of topological approach is crucial, and, in particular, an algebraic topological method known as the homological Conley index theory provides a powerful tool, especially that this index can be efficiently computed with the publically available CHomP software package [8] which is based on the theory introduced in [20,29,36] (see also references therein).…”