A fixed point approach to homological perturbation theory

Abstract

Abstract.We show that the problem addressed by classical homological perturbation theory can be reformulated as a fixed point problem leading to new insights into the nature of its solutions. We show, under mild conditions, that the solution is essentially unique.

Abstract Hodge Decomposition and Minimal Models for Cyclic Algebras

“…[2]. It is clear that there is a direct sum decomposition V = Ker t ⊕ Im t into orthogonal sub dg vector spaces; moreover s determines a contractible homotopy on Ker t. The Hodge decomposition is then trivial if and only if Im t = V , and harmonious if and only if Im t has zero differential and thus carries the homology of V .…”

Transferring algebra structures up to homology equivalence

“…This method for constructing τ and ∂ uses the perturbation lemma [6], [11], [2] in the following way. First one applies the free tensor coalgebra functor T c (·) to (4).…”

“…Consider the SDR (4) and suppose that A is an algebra and let t be the initiator (in the language of [2]) for the SDR (9). We begin with an algebraic observation that depends on the side conditions (5).…”

“…Given a simply connected coalgebra C and a module M, suppose that < π k (φ(c) (1) ) ⊗ π n−k (φ(c) (2) )>.…”

“…We introduce the non-coassociative operation φ on C and write c <1> ⊗ c <2> = φ(c) (1) ⊗ φ(c) (2) to see that the first few terms of the generalized GM-formula are Dual to our observation in section (3.1), we see that these terms are made up from obstructions to the non-coassociativity of φ. Once again, the point of the main theorem is that these higher obstructions are exactly reproduced in the tensor trick and the double induction.…”