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# Homology, Homotopy and Applications

## Volume 13 (2011)

### Number 2

### L-infinity maps and twistings

Pages: 175 – 195

DOI: https://dx.doi.org/10.4310/HHA.2011.v13.n2.a12

#### Authors

#### Abstract

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

#### Keywords

differential graded Lie algebra; Maurer-Cartan element; $A_\infty$ algebra; graph homology; Morita equivalence

#### 2010 Mathematics Subject Classification

16E45, 18D50, 57T30, 81T18

Published 25 January 2012