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.