Algorithms in algebraic topology and homological algebra: the problem of the complexity

This survey tackles the problem of the high computational complexity lying in most of the algorithms in Algebraic Topology and Homological Algebra. We deal with three particular algorithms: the computation of the homology of commutative differential graded algebras, the homology of principal twisted cartesian products of Eilenberg–Mac Lane spaces and a combinatorial method computing Steenrod Squares.