The aim of Homological Perturbation Theory is to construct small chain complexes from large ones. It was originally developed for the calculation of chain models of the total spaces of fiber bundles, but has since developed into a useful general computational tool.
for all . The Homological Perturbation Lemma states that can be given a new differential such that there is a quasi-isomorphism .
The main point is that there is an explicit formula for the new chain homotopy as
There is considerable interest in describing the new differential in terms of a twisting cochain.
This result derived from earlier work of G. Hirsch, E.H. Brown, Weishu Shih, and has been widely developed into a useful theoretical and computational tool by Guggenheim, Lambe, Stasheff and others.
Homological perturbation theory is a key tool in the construction of BRST-BV complexes, where the quantum BV complex is a perturbation of a classical BV-complex. See (Gwilliam, section 2.5). In this context Wick's lemma in quantum field theory is a direct consequence of the homological perturbation lemma (Gwilliam, section 2.5.2).
Other references include
Donald W. Barnes, Larry A. Lambe, A fixed point approach to homological perturbation theory Proc. Amer. Math. Soc. 112 (1991), no. 3, 881–892.
For applications to “make computable” a bicategory of isolated hypersurface singularities and matrix factorisations comp that has been studied in the context of topological field theory, using the formulation in (Barnes-Lambe 91) is in
See also linear logic.
V. Álvarez , J.A. Armario , P. Real , B. Silva , Homological Perturbation Theory And Computability Of Hochschild And Cyclic Homologies Of Cdgas (pdf)
Discussion in the context of BV-quantization is in section 2.5 of