[An excellent set of notes by Sergeraert.]
Computational homology, by Kaczynski, Mischaikow, Mrozek. Springer
http://mathoverflow.net/questions/57166/computational-software-in-algebraic-topology
http://mathoverflow.net/questions/22232/which-properties-of-finite-simplicial-sets-can-be-computed
arXiv:1208.3816 Constructive Homological Algebra and Applications from arXiv Front: math.KT by Julio Rubio, Francis Sergeraert This text was written and used for a MAP Summer School at the University of Genova, August 28 to September 2, 2006. Available since then on the web site of the second author, it has been used and referenced by several colleagues working in Commutative Algebra and Algebraic Topology. To make safer such references, it was suggested to place it on the Arxiv repository
It is a relatively detailed exposition of the use of the Basic Perturbation Lemma to make constructive Homological Algebra (standard Homological Algebra is not constructive) and how this technology can be used in Commutative Algebra (Koszul complexes) and Algebraic Topology (effective versions of spectral sequences).
arXiv:1206.4345 Geometric Objects and Cohomology Operations from arXiv Front: math.AT by Rocio Gonzalez-Diaz, Pedro Real Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it
In the literature, there exist various algorithms for computing the homology groups of simplicial complexes but concerning the algorithmic treatment of cohomology operations, very little is known. In this paper, we establish a version of the incremental algorithm for computing homology which saves algebraic information, allowing us the computation of the cup product and the effective evaluation of the primary and secondary cohomology operations on the cohomology of a finite simplicial complex. We study the computational complexity of these processes and a program in Mathematica for cohomology computations is presented.
nLab page on Computational algebraic topology