# On the Goldman-Millson theorem for A-infinity algebras in arbitrary characteristic

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## Authors

Milham, Alexander

## Issue Date

2023

## Type

Dissertation

## Language

## Keywords

A-infinity algebras , Homotopical algebra , Homotopy theory , Maurer-Cartan simplicial set

## Alternative Title

## Abstract

Simplicial sets provide a combinatorial model for the homotopy category of topological spaces. Arich source of simplicial sets comes from algebra. A classical case is the Dold-Kan correspondence
between chain complexes and simplical abelian groups. Under this correspondence, quasiisomorphisms
of chain complexes are mapped to homotopy equivalences of simplicial abelian groups
and the homology groups of a chain complex are isomorphic to the homotopy groups of the corresponding
simplicial abelian group. Another example is the nerve of a category, which sends a small
category to a simplicial set.In the same spirit, this thesis is about constructing simplicial sets from A-infinity algebras over a fieldof arbitrary characteristic. A-infinity algebras are the homotopy theoretic analog of differential graded
(dg) associative algebras. First we leverage some abstract homotopical algebra to prove that the
nerve functor, which assigns a simplicial set to a filtered A-infinity algebra A, sends filtered quasi-isomorphisms
to homotopy equivalences. This is an analog of a classical theorem of W. Goldman
and J. Millson concerning dg Lie algebras in characteristic zero. Next, we characterize the homotopy
groups of the nerve of A in terms of the cohomology algebra H(A) and its group of quasi-invertible
elements. Finally, as an application, we show that in the characteristic zero case, the nerve of A
is homotopy equivalent to the simplicial Maurer-Cartan set of its commutator L-infinity algebra. This
answers a question posed in a recent publication by N. de Klein and F. Wierstra.