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# Homology, Homotopy and Applications

## Volume 2 (2000)

### Number 1

### Homological perturbation theory and associativity

Pages: 51 – 88

DOI: http://dx.doi.org/10.4310/HHA.2000.v2.n1.a5

#### Author

#### Abstract

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.

#### Keywords

filtered algebra, graded algebra, resolutions, homological perturbation, differential homological algebra, augmented algebra, chain complex, contraction

#### 2010 Mathematics Subject Classification

18G10, 18Gxx, 55U15, 55Uxx