A relative version of the ordinary perturbation lemma

The perturbation lemma and the homotopy transfer for L-infinity algebras is proved in a elementary way by using a relative version of the ordinary perturbation lemma for chain complexes and the coalgebra perturbation lemma.

💡 Research Summary

The paper presents a new “relative” formulation of the ordinary perturbation lemma (OPL) for chain complexes and shows how this formulation yields elementary proofs of the perturbation lemma and the homotopy transfer theorem for (L_{\infty})-algebras. The authors begin by recalling the classical OPL, which requires a global contraction ((i,p,h)) on a chain complex ((C,d)). In many applications one wishes to keep a distinguished subcomplex (A\subset C) untouched by the perturbation; the standard OPL does not address this situation directly. To fill the gap, the authors introduce the notion of a relative contraction: a triple of maps (i\colon A\to C), (p\colon C\to A), and a homotopy (h\colon C\to C) satisfying (p,i=\mathrm{id}_A) and (dh+hd=\mathrm{id}_C-i,p), together with the extra condition that the perturbation (\delta) vanishes on (A) (i.e. (\delta\circ i=0) and (p\circ\delta=0)). Under these hypotheses they prove the Relative Ordinary Perturbation Lemma: the perturbed differential (d’ = d+\delta) admits modified maps \