# L-infinity maps and twistings

## TL;DR

This paper constructs an L-infinity map into the truncated Chevalley-Eilenberg complex and its analogues, with applications to deformation theory and graph homology, using Maurer-Cartan functors and twistings.

## Contribution

It introduces a new L-infinity map into truncated complexes and explores its role in homotopy fiber sequences, deformation theory, and graph homology.

## Key findings

- Constructed an L-infinity map to truncated Chevalley-Eilenberg complexes.
- Established a homotopy fiber sequence involving these complexes.
- Applied the framework to deformation theory and graph homology.

## Abstract

We give a construction of an L-infinity map from any L-infinity algebra into its truncated Chevalley-Eilenberg complex as well as its cyclic and A-infinity analogues. This map fits with the inclusion into the full Chevalley-Eilenberg complex (or its respective analogues) to form a homotopy fiber sequence of L-infinity-algebras. Application to deformation theory and graph homology are given. We employ the machinery of Maurer-Cartan functors in L-infinity and A-infinity algebras and associated twistings which should be of independent interest.