As a first unknown (kinda, sorta, it still falls under the category of path algebra quotients treated by Keller) A_{∞}-calculation, I shall find the A_{∞}-structure of .

To do this, I fix the group algebra

and the cohomology ring

with ,

Furthermore, we pick a canonical nice resolution P, continuing the one I calculated previously. This has the i:th component Λ^{i+1}, and the differentials looking like

for differentials starting in odd degree, and

for differentials starting in even degree. The first few you can see on the previous calculation, or if you don’t want to bother, they are

Now, armed with this, we can get cracking. By lifting, we get canonical representating chain maps for x,y,z described, loosly, by the following:

x takes an element in , keeps the first, third, et.c. elements and throws out the even ordered elements; so

For an element in , the last element gets extra treatment, so

For the lowest degrees, we also have

This describes the image of our quasi-isomorphism .

For , we get the interleaved effect: every second element, but now with even indices, and some extra treatments at the end. So an element in will behave like

and for an element in we get

The last generator, z, is rather boring. The corresponding chain map only lifts elements to other degrees: shaving off the first two components of whatever it is applied to.

We define on products of the generators by simply composing the corresponding chain maps as long as the product is defined. The interesting stuff, from an A_{∞} point of view occurs when the product vanishes, thus for x,y in the first line. A calculation shows us that xy is the only interesting element of the ideal , since , and we define and .

Thus, we’d be curious as to what happens with xy, and yx. Both products are zero in the cohomology ring; but the composition of the corresponding chain maps are not zero.

By calculation, we get given in the first few step by the matrices

and so on, with the lower right entry alternatingly ab and ba.

is the same thing, but with ab and ba interchanged.

We want and to be homotopies between the 0 chain maps, and these two respectively. By juggling the relevant matrices in Magma for a while, I conclude that has lower right entry a for all odd degree map components, and has the entry above that b, same components. All even components vanish.

Thus, for the first nonzero component, we have

for x,y and

for y,x.

Now, if we look at Φ_{3}, most of the possibilities vanish because of the way we defined f_{2} for the non xy, yx cases. Thus, the only interesting entries remaining are xyx and yxy. These are, respectively,

and calculation of these expressions is a matter of composing the chain maps we have a tentative grasp of already. This gives us the case of xyx alternatingly between b in lower right corner for all even degrees and a just to the left of it for odd degrees, thus starting with the matrices

and for the case yxy, we have a similar pattern, but with a in the lower right for odd degrees and b above it for even degrees, giving the first two maps as

So, we can immediately conclude that our m_{3} is going to be the zero element of Γ, since none of these chain maps are homotopic to any non-trivial coclass representatives. Thus we’ll need to find homotopies from these to the zero chain maps for our values of f_{3}.

A homotopy suitable for f_{3}(x,y,x) would be h with

It goes on, but I have not yet discerned a pattern clear enough to describe a generic element of this chain map. I probably should, at some point.

For f_{3}(y,x,y), we would need a homotopy for the other sequence, and calculations lead me to put down h with

and here we can discern a pattern to the matrices. They will have a sequence of 1 starting out at the third column, and going down skipping every second place.

Armed with these calculations, we may set out to calculate for our pleasure Φ_{4}. Due to the heuristic we use in defining the f_{i} for things “lifted” from the generators we’ve defined above, we shall discard anything except for xyxy and yxyx from study; all other cases will just give the relations we use in calculating f_{2} or f_{3} from the cases we’ve calculated.

Thus, we’ll be interested in

and

and thus we can, by using the already calculated matrices and a CAS (I use Magma right now) just calculate the matrices. In both these cases we get the same thing: the matrices

and so on; which are precisely the matrices we chose for our .

Thus, we’ll set m_{4}(x,y,x,y)=m_{4}(y,x,y,x)=z and f_{4}=0, and stop calculating right here.

[…] have previously calculated the A∞-structure for the cohomology ring of D8. Now, while trying to figure out how to make my work continue from here, I tried working out what […]