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\begin{document}

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\section*{homological perturbation theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{homological_perturbation_lemma}{Homological perturbation lemma}\dotfill \pageref*{homological_perturbation_lemma} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{BVComplexesAndWickLemma}{BV-complexes and Wick's lemma}\dotfill \pageref*{BVComplexesAndWickLemma} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The aim of \emph{Homological Perturbation Theory} is to construct small [[chain complex]]es 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.

\hypertarget{homological_perturbation_lemma}{}\subsection*{{Homological perturbation lemma}}\label{homological_perturbation_lemma}

Let $(X,d), (Y,d)$ be [[chain complexes]] over a [[commutative ring]] $R$ and let $f: X \to Y, \nabla: Y \to X$ be [[chain maps]], and $\Phi: X \to X$ be a [[chain homotopy]] such that

\begin{displaymath}
f \nabla=1, \quad \nabla f= 1 + d\Phi + \phi d,
\end{displaymath}
\begin{displaymath}
f\Phi = 0, \Phi \nabla=0, \Phi^2=0, \Phi d \Phi= - \Phi.
\end{displaymath}
Let $X,Y$ have [[filtrations]] $F^*$ bounded below by $0$ and preserved by $\nabla,f, \Phi$ and the [[differentials]] on $X,Y$. Suppose $X$ has another differential $d^\tau$ with the property that

\begin{displaymath}
(d^\tau -d)F^p X \subseteq F^{p-1} X
\end{displaymath}
for all $p \geq 0$. The \textbf{Homological Perturbation Lemma} states that $Y$ can be given a new differential $d^\tau$ such that there is a [[quasi-isomorphism]] $(Y, d^\tau) \to (X, d^\tau)$.

The main point is that there is an explicit formula for the new chain homotopy as

\begin{displaymath}
\Phi^\tau= \sum _{r=0}^\infty \Phi (1+ d^\tau \Phi)^r.
\end{displaymath}
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.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{BVComplexesAndWickLemma}{}\subsubsection*{{BV-complexes and Wick's lemma}}\label{BVComplexesAndWickLemma}

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 (\hyperlink{Gwilliam}{Gwilliam, section 2.5}). In this context [[Wick's lemma]] in [[quantum field theory]] is a direct consequence of the homological perturbation lemma (\hyperlink{Gwilliam}{Gwilliam, section 2.5.2}).

\hypertarget{references}{}\subsection*{{References}}\label{references}

Review includes

\begin{itemize}%
\item [[Marius Crainic]], \emph{On the perturbation lemma, and deformations} (\href{http://arxiv.org/abs/math/0403266}{arXiv:math/0403266})


\item [[Johannes Huebschmann]], \emph{A survey on homological perturbation theory} (\href{http://math.univ-lille1.fr/~huebschm/data/talks/courant.pdf}{pdf})



\end{itemize}
Other references include

\begin{itemize}%
\item [[Ronnie Brown]], \emph{The twisted Eilenberg-Zilber Theorem}, Simposio di Topologia (Messina, 1964) pp. 33--37 Edizioni Oderisi, Gubbio. (\href{http://pages.bangor.ac.uk/~mas010/pdffiles/twistedez.pdf}{pdf})


\item Donald W. Barnes, [[Larry Lambe|Larry A. Lambe]], \emph{A fixed point approach to homological perturbation theory} Proc. Amer. Math. Soc. 112 (1991), no. 3, 881--892.



\end{itemize}
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 (\hyperlink{BarnesLambe91}{Barnes-Lambe 91}) is in

\begin{itemize}%
\item [[Daniel Murfet]], \emph{Computing with cut systems}, (\href{http://arxiv.org/abs/1402.4541}{arXiv:1402.4541})

\end{itemize}
See also \emph{[[linear logic]]}.

Discussion with an eye towards [[Hochschild cohomology]] and [[cyclic cohomology]] is in

\begin{itemize}%
\item [[Larry Lambe|Larry A. Lambe]], \emph{Homological Perturbation Theory Hochschild Homology and Formal Groups} Cont. Math., vol 189, AMS, 1992 (\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.64.8783&rep=rep1&type=pdf}{pdf})


\item V. \'A{}lvarez , J.A. Armario , P. Real , B. Silva , \emph{Homological Perturbation Theory And Computability Of Hochschild And Cyclic Homologies Of Cdgas} (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.6955}{pdf})



\end{itemize}
Discussion in the context of [[BV-quantization]] is in section 2.5 of

\begin{itemize}%
\item [[Owen Gwilliam]], \emph{Factorization algebras and free field theories} PhD thesis (\href{http://math.berkeley.edu/~gwilliam/thesis.pdf}{pdf})

\end{itemize}
[[!redirects homological perturbation lemma]]



\end{document}