An Entry From Encyclopaedia of Mathematics Supplement

Kluwer Academic Publishers 2000 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 ∇ = 1 X (the identity map on X) and D(φ) = 1 Y − ∇f (cf. also Complex (in homological algebra)). Here it is assumed that the differentials d X and d Y of X and Y respectively are of degree −1, the degree of φ is +1 and D(φ) = d Y φ + φd Y , i.e. φ is a chain homotopy between the identity and ∇f , while f ∇ is the identity. A standard notation for an SDR is the following: (1) (X ∇ − −− → ←−− f Y, φ). The notion of an SDR is essentially equivalent to what is called a contraction in [5]. Side Conditions: There are three additional conditions for an SDR which are needed to achieve both theoretical and computational results. They are called the side conditions: φ∇ = 0, f φ = 0, and φφ = 0. Fortunately, these may always be satisfied as follows: if the first two conditions do not hold, replace φ by ¯ φ = D(φ)φD(φ), then the new data given by ∇, f , and ¯ φ defines an SDR in which the first two conditions hold. If the third condition does not hold, and the first two do, replace ¯ φ by Φ = ¯ φd ¯ φ and the new data given by ∇, f , and Φ defines an SDR in which all conditions hold [17].