Excision-preserving cubical approach to the algorithmic computation of the discrete Conley index

Abstract

We introduce a new approach to the algorithmic computation of the Conley index for continuous maps. We use the technique of splitting an index pair into two layers which is inspired by the work of Mrozek, Reineck and Srzednicki [M. Mrozek, J.F. Reineck, R. Srzednicki, The Conley index over a base, Trans. Amer. Math. Soc. 352 (2000) 4171–4194]. The main advantage of our construction over the approach based directly on the one introduced by Mischaikow, Mrozek and Pilarczyk [K. Mischaikow, M. Mrozek, P. Pilarczyk, Graph approach to the computation of the homology of continuous maps, Found. Comput. Math. 5 (2005) 199–229] is that our cubical sets have the excision property. Moreover, our solution has some advantages in comparison to the approach recently proposed by Mrozek [M. Mrozek, Index pair algorithms, Found. Comput. Math. 6 (2006) 457–493].