In this paper, we prove various results concerning DGA-algebras in the context of the Homological Perturbation Theory. We distinguish two class of contractions for algebras: full algebra contractions and semi-full algebra contractions. A full algebra contraction is, in particular, a semi-full algebra contraction. Taking a full algebra contraction and an "algebra perturbation" as data of the Basic Perturbation Lemma, the Algebra Perturbation Lemma (or simply, F-APL) of [20] and [29] appears in a natural way. We establish here a perturbation machinery, the Semi-Full Algebra Perturbation Lemma (or, simply, SFAPL) that is a generalization of the previous one in the sense that the application range of SF-APL is wider than that of F-APL. We show four important applications in which this result is essential for the construction of algebra or coalgebra structures in various chain complexes. 1. Introduction Homological Perturbation Theory [52, 14, 8, 9, 18, 36, 19, 20, 29] is a set of technique...

Working in the framework of the Simplicial Topology, a method for calculating the p-local homology of a twisted cartesian product X(#, m, #, # # , n) = K(#, m) # K(# # , n) of Eilenberg-Mac Lane spaces is given. The chief technique is the construction of an explicit homotopy equivalence between the normalized chain complex of X and a free DGA-module of finite type M , via homological perturbation. If X is a commutative simplicial group (being its inner product the natural one of the cartesian product of K(#, m) and K(# # , n)), then M is a DGA-algebra. Finally, in the special case K(#, 1) ## X K(# # , n), we prove that M can be a small twisted tensor product. 1

We reinterpret the classical theory of cocyclic operations in terms of permutations and homotopy equivalences of explicit chains. The essential tools we use are Homological Perturbation Theory and Eilenberg-Zilber Theorem. The main objective of this technique is the final identification of cohomology operations at cochain level.

Analyzing the transference of the coalgebra structure on the homology of CDGAs

We deal with the problem of obtaining explicit simplicial formulae defining the classical Adem cohomology operations at the cochain level. Having these formulae at hand, we design an algorithm for computing these operations for any finite simplicial set.

We present here a combinatorial method for computing cup-i products and Steenrod squares of a simplicial set X. This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of commutativity of the cup product on the cochain level) in terms of face operators of X. A generalization of this method to Steenrod reduced powers is sketched.