The Infinite Seminar

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.

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.