The Infinite Seminar

Ben Webster asked, in a comment to the post Question for the audience for pointers to literature on the computation of $A_{\mathrm{\infty }}$-structures. Since I’m still out traveling, and far from a university, I won’t give any real pointers, but rather stick to namedropping. It’s not very difficult, using the names, to dig out the relevant article references from MathSciNet.

I also suffer from being slightly new to the field. There are people out there with a much better overview of what has been done, and what does apply. (Jim Stasheff, I’m looking at you! :))

At the core of $A_{\mathrm{\infty }}$-structures on Ext algebras lies the so called minimality theorem, proven by A Whole Range Of People. It states, roughly, that if $A$ is a dg-algebra, then the $A_{\mathrm{\infty }}$-structure on $A$ given by $m_1=d$ and $m_2=\cdot$ induces an $A_{\mathrm{\infty }}$-structure on $H^{*}A$, together with a quasiisomorphism $H^{*}A\to A$ in such a way that any two such induced structures are quasiisomorphic to each other.

And with $A$ taken as, for instance, ${\mathrm{End}}_A(\mathrm{pS})$, for $\mathrm{pS}$ suitable resolution (i.e. dg-module quasiisomorphic to the $A$-module $S$), we can view ${\mathrm{Ext}}_A(S,S)=H^{*}({\mathrm{End}}_A(\mathrm{pS}))$, and thus find an $A_{\mathrm{\infty }}$-structure on Ext induced by the obvious structure on the endomorphism dg-algebra of a chain complex.

Now, the trick is how to get such a structure. One of the earliest mentions I know of is the paper by Kadeishvili, where a purely algorithmic approach is taken. He gives a recursion, using the Stasheff axioms, where, in order to calculate a certain higher product for a certain input, you calculate a sum of products and compositions of lower products, ending up with something that is, inductively, known. This way, you end up with calculation of specific products reducing to a matter of taking preimages under certain differentials and a lot of bookkeeping.

This method is the one I’m basically talking about in my previous post.

Another method floating around uses Homotopy Perturbation Theory (or was it Homology Perturbation Theory - I keep forgetting). Some of the relevant names here are Huebschmann, Stasheff, Gugenheim, Johansson (Lennart, not Mikael!!!) and Lambe. The idea here is to find a strong deformation retract $A\to H^{*}A$, and work with the portions of that to find everything you need.

This kind of approach I have seen used by Berciano in her work with the KENZO module ARAIA-CRAIC for calculation of $A_{\mathrm{\infty }}$-coalgebra structures in topological contexts, and is also being used by Berciano-Umble in at least one recent preprint.

Finally, Merkulov has taken the basic idea of HPT and refined it for cases where a lot is already known. He requires a vector space splitting of the dg-algebra $A=H\oplus B\oplus N$, where $H=H^{*}A$, $H\oplus B=\ker d$ and $N$ is the complement of $H\oplus B$. Using this, and a few functions - a projection $\pi$ down on $H$ and a homotopy $\pi \sim \mathrm{Id}$, he gets enough data to be able to write down the $A_{\mathrm{\infty }}$-structure on $H$ with each function and each component of the quasiisomorphism being given by sums of planar evaluation trees, with each internal edge being an application of the homotopy, and each vertex being a normal multiplication in the dg-algebra.

It has the good property of being generic and at the same time very explicit. However, unless this splitting is given, and the homotopy found, you can’t do very much with it.

I hope I haven’t now missed anyone important in the overview. Please correct me if I have.