A-infinity structures

From Talbot

Get PDF | Get log | Regenerate PDF | Export Source

---

\documentclass{article}
\usepackage{amsmath,amsfonts,amssymb,epsfig,color}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
 
\newcommand{\iso}{\cong}
\parskip1em
\parindent0em
\begin{document}
 
\underline{$A_\infty$-structures}
 
This talk introduces $A_\infty$-algebras. I want to keep things very 
concrete, stating some of the main constructions explicitly (as opposed
to, let's say, describing them as features of the model category of
$A_\infty$-algebras). The main point is how much information is actually
contained in the higher order products. There are no strict prerequisites
for
giving this talk, but background in rational homotopy theory may be
helpful.
 
\underline{Plan}
 
{\it Definitions.} It will be enough for us to work over a field, for
instance $\C$. Define the notion of $A_\infty$-algebra, homomorphism
of $A_\infty$-algebras, and quasi-isomorphism. The main theorem is
that any quasi-isomorphism is invertible (proof will follow later on).
Basic references are [Keller, lectures on $A_\infty$-algebras and
modules], [Seidel, Fukaya categories and Picard-Lefschetz theory,
Chapter 1], [Lefevre-Hasegawa, sur les $A_\infty$-cat\'egories, it's
a thesis but I have a copy].
 
At some point, one needs to explain that an $A_\infty$-structure
yields triple Massey products on the cohomology level. This is nice
because one can see the $m_3$ term appearing explicitly in the
formula for the Massey product (it is also a kind of baby version
of the next topic).
 
{\it Perturbation theory.} Let's see the proof of the Perturbation
Lemma. I strongly suggest following the explicit sum-over-trees method
given in
[Kontsevich-Soibelman, Homological mirror symmetry and torus fibrations].
Other references are [Markl, Transferring $A_\infty$ (strongly homotopy
associative) 
structures] or [Merkulov, Strongly homotopy algebra of a K{\"a}hler
manifold]. 
 
The Perturbation Lemma implies the theorem above on invertibility
of quasi-isomorphisms. Compare the corresponding argument for the
$L_\infty$-case in [Kontsevich, Deformation quantization of Poisson
manifolds].
 
{\it Examples.} As an application of the Perturbation Lemma, the
cohomology of any space has an $A_\infty$-structure (unique up to
isomorphism, but not unique on the nose). A classical theorem of
Deligne-Griffiths-Morgan-Sullivan says that for compact K{\"a}hler
manifolds, the $A_\infty$-structure is in fact trivial (the 
original paper does not use $A_\infty$-terminology, but proves
an equivalent result; for an $A_\infty$ viewpoint, see the more
recent [Merkulov, op. cit.], which unfortunately doesn't quite
get to the theorem).
 
Another topological example which I quite like is the following one. Let
$M$ be a homology three-sphere. We consider the moduli space of flat
$GL(N)$
connections over $M$. The ordinary cohomology describes the Zariski
tangent space to that moduli space (at the trivial connection), but the 
$A_\infty$-structure on it describes the entire formal neighbourhood
of the trivial connection. This involves rewriting the Maurer-Cartan
equation on cochains, $da + a^2 = 0$, in terms of the $A_\infty$-structure
as generalized Maurer-Cartan equation on cohomology [no explicit
reference,
but old papers of Goldman and Millson on moduli spaces of flat connections
will be helpful].
 
{\it Dependencies.} None! However, we have a talk later on on Hochschild
cohomology, which will complement this one in some respects.
\end{document}