An Entry From Encyclopaedia of Mathematics Supplement II Kluwer Academic Publishers

Homological Perturbation Theory – A theory that concerns itself with of a collection of techniques for deriving chain complexes which are both smaller and chain homotopy equivalent to a given chain complex (cf. also Complex (in homological algebra)). It is motivated by the desire to find effective algorithms in homological algebra. The cornerstone of the theory is an important algorithm which, when convergent, is commonly called the ‘perturbation lemma’. To understand the statement of the perturbation lemma, some preliminary notation is needed. Strong Deformation Retraction Data: It will be assumed that R is a commutative ring with unit and that all chain complexes are over R are free (cf. also Simplicial complex). A strong deformation retract from X to Y consists of two chain complexes X and Y, such that there are chain maps ∇ : X → Y, f: Y → X, and a chain homotopy φ: Y → Y such that f ∇ = 1X (the identity map on X) and D(φ) = 1Y − ∇f (cf. also Complex (in homological algebra)). Here it is assumed that the differentials dX and dY of X and Y respectively are of degree −1, the degree of φ is +1 and D(φ) = dY φ + φdY, i.e. φ is a chain homotopy between the identity and ∇f, while f ∇ is the identity. A standar