The Infinite Seminar

In the arXiv mailing from today, February 2nd 2008, there was a large chunk of $A_{\mathrm{\infty }}$-related preprints from Lyubashenko and Manzyuk. Most of these preprints are getting published in journals, or in Max-Planck-Institute report series.

For reference, the arXiv posts I’ll be talking about here are:
http://arxiv.org/abs/0802.2885
http://arxiv.org/abs/math/0211037
http://arxiv.org/abs/math/0306018
http://arxiv.org/abs/math/0312339
http://arxiv.org/abs/math/0701165

In general sweeps - acquired from scanning the abstracts - Lyubashenko and Manzyuk are talking about $A_{\mathrm{\infty }}$-categories as defined by Fukaya, and doing various basic constructions with them - demonstrating that various definitions of unitality coincide, showing a relation to Serre k-linear functors, constructing quotient categories with the same kind of structure et.c.

Now, one question that occurs to me quickly is the following:
Could an $A_{\mathrm{\infty }}$-algebra be considered to be a one-object $A_{\mathrm{\infty }}$-category? And if this is so, can the quotient constructions Lyubashenko and Manzyuk are using be extended (possibly significantly) to form a transferral of $A_{\mathrm{\infty }}$-(co)algebra structure across surjections in general?

And the way off target question: Does this somehow give us a way to transfer $A_{\mathrm{\infty }}$-(co)algebra structures along things like the restriction map in group (co)homology?

In my doctoral thesis (soon to be released to the general public), I spend some time discussing a blackbox technique for computing A-infinity structures on Ext algebras using Kadeishvili’s proof of the minimality theorem as an inspiration for an algorithmic approach.

Basically, we build up a quasi-isomorphism from $H^{*}A$ to $A$ by only ever defining the bits and pieces we really need to define, and figuring out what they should be by staring at the Stasheff morphism axioms and recursing down until everything we need to know is computed.

During my visit to Sydney last autumn, I implemented this as a module for the computer algebra system MAGMA. Now, as a part of my last minute thesis revisions, I have tracked down and fixed (hopefully all) bugs in the computation. Hence, beginning with release 2.14-10 of MAGMA, there will be an A-infinity computation module distributed with the system which at least computes the already known A-infinity structures on $H^{*}(C_{p^n},Z/p)$ correctly. It can also be used for explorations in p-group cohomology.

Some examples (the examples are run in MAGMA 2.14-9, but with a development implementation of the A-infinity module):

> G := CyclicGroup(3);
> Aoo := AInfinityRecord(G,10);
> S<x,y> := Aoo`S;
> HighProduct(Aoo,[ x : i in [1..10]]);
0
> { #k : k in Keys(Aoo`m) | not IsZero(HighProduct(Aoo,k)) };
{ 2, 3 }
> { #k : k in Keys(Aoo`m) | not IsZero(HighMap(Aoo,k)) };
{ 2 }

This example shows us that the only higher products and quasi-isomorphism components that do not vanish in $H^{*}(C_3,Z/3)$ have arity 2 and 3. With some extra verification, we can use some of the latest arguments in my thesis to convince ourselves that the entire A-infinity structure on this cohomology ring follows from this particular computation.

> G := DihedralGroup(4);
> Aoo := AInfinityRecord(G,10);
> S<x,y,z> := Aoo`S;
> HighProduct(Aoo,[x,y,x,y]);
z
> HighProduct(Aoo,[x,y,x,y^2]);
y*z
> HighProduct(Aoo,[x,y,x,y^3]);
y^2*z
> HighProduct(Aoo,[x,y,x,y^4]);
y^3*z

These are essentially a small, but representative subset of the computations on the cohomology of the dihedral group with 8 elements that I presented in T’bilisi in 2006.

So - if you have a MAGMA installation, 2.14-x, and want to play with this before 2.14-10 is released, prod me and we’ll work something out.

arXiv:0711.4499
Steffen Sagave - DG-algebras and derived A-infinity algebras

In this paper, the theory of $A_{\mathrm{\infty }}$-algebras as sketched in Kellers survey papers is developed to give bigraded (homological and “original”) resolutions of dg-algebras, and construct minimal models for dg-algebras using an extended notion of $A_{\mathrm{\infty }}$-algebras called derived $A_{\mathrm{\infty }}$-algebras. These take a bigraded structure into account.

Sagave proves a derived analogue of the minimality theorem applicable for any dg-algebra over any commutative ring.

arXiv:0712.0996

Valery A. Lunts: Formality of DG algebras (after Kaledin)

This paper, which states in the arXiv summary that it’s an early draft, and welcomes additional input, sets out to develop the background in order to prove a result stated in a paper by Dima Kaledin. Kaledin picks out a special cohomology class in a Hochschild cohomology ring, which acts as an obstruction to formality of the algebra studied. The core results deals with how this obstruction is realized in the Kaledin class and how formality of $A_{\mathrm{\infty }}$-algebras gets inherited in various constructions.

There is a paper by Loday on the arXiv since about a month ago, entitled The diagonal of the Stasheff polytope. The basic idea of the paper is to introduce a new operad ${\mathrm{AA}}_{\mathrm{\infty }}$, built on the simplicial chains of Lodays triangulation of the associahedron, and using the relative simplicity of forming diagonals on simplicial complexes to generate a reasonably natural diagonal on this new operad.

With quasiisomorphisms from our familiar $A_{\mathrm{\infty }}$-operad to ${\mathrm{AA}}_{\mathrm{\infty }}$ and back again, he then constructs a diagonal on $A_{\mathrm{\infty }}$, formed by going to ${\mathrm{AA}}_{\mathrm{\infty }}$ and computing a simplicial diagonal, which finally gets deformed into a diagonal on $A_{\mathrm{\infty }}$.

Thus, this construction is based in a slightly different approach to diagonal computation than both the Saneblidze-Umble construction and the Markl-Schnider paper; but Loday conjectures equality between the two constructions based on a comparison of the results for the diagonal 5-ary operation.

If anyone out there has read more of the paper than I have (or if Loday himself is reading this sporadic blog), I would appreciate some nudges on how to internalize the deformation enough to fix it in computer code.

towards-relative-symplectic-field-theory-report.pdf

A sneak preview of my current project — which will end up as about a third of my PhD thesis:

dynkin:~/magma> magma
Magma V2.14-D250907   Wed Sep 26 2007 13:19:51 on dynkin   [Seed = 1]
Type ? for help.  Type -D to quit.

Loading startup file “/home/mik/.magmarc”

> Attach(”homotopy.m”);
> Attach(”assoc.m”);
> Aoo := ConstructAooRecord(DihedralGroup(4),10);
> S := CohomologyRingQuotient(Aoo`R);
> CalculateHighProduct(Aoo,[x,y,x,y]);
z
> exit;
Total time: 203.039 seconds, Total memory usage: 146.18MB

AJ Tolland links to some seminar notes he took from a seminar by Kevin Costello on $A_{\mathrm{\infty }}$-algebras in topological string theory.

Thanks to AJ for telling me what I got wrong in the first take.

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.

The blog is in quite a summer lull, and I’m not doing near my share of keeping it up. I’m blaming my vacation and my marriage ceremony tomorrow.

Anyway, here is a question I was thinking about on my way home. About appropriate terminology for my own research. I’m putting quite a bit of thought into calculation of $A_{\mathrm{\infty }}$-structures on Ext-algebras in cases where the usual calculation methods - Homotopy perturbation theory, Merkulov’s method, et.c. - do not really work well since the required data about the endomorphism ring of the appropriate chain complex ends up being much too large. So far I have been calling this local computation, but it struck me that it might end up confusing those more used to local being used for .. say .. localization in various contexts.

On the way I thought about blind computation, to indicate the lack of information compared to the more global methods, but this doesn’t seem to be quite it either.

Thus a question to the readers (and writers?) of this blog: what would be a good word to describe my particular brand of computation of $A_{\mathrm{\infty }}$-structures on $H_{*}(A)$, for $A$ a DG-algebra which gets viewed as a black box, capable of performing calculations, but not of displaying its internals in any good way?

Some of the recent preprints seen in the arXiv mailings include:

Mikael Johansson: A partial $A_{\mathrm{\infty }}$-structure on the cohomology of $C_n\times C_m$

Suppose k is a field of characteristic 2, and n,m≥2 powers of 2. Then the $A_{\mathrm{\infty }}$-structure of the group cohomology algebras $H^{*}(C_n,k)$ and $H^{*}(C_m,k)$ are well known. We give results characterizing an $A_{\mathrm{\infty }}$-structure on $H^{*}(C_n\times C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.

Alastair Hamilton & Andrey Lazarev: Cohomology theories for homotopy algebras and noncommutative geometry

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely $A_{\mathrm{\infty }},C_{\mathrm{\infty }}$ and $L_{\mathrm{\infty }}$-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of $C_{\mathrm{\infty }}$-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.

Alastair Hamilton & Andrey Lazarev: Symplectic $A_{\mathrm{\infty }}$-algebras and string topology operations

In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincar\’e duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading `string topology’. Our method is based on obstruction theory for $C_{\mathrm{\infty }}$-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.

Yes, this post includes a bit of shameless self-promotion.