# An A∞-structure on the cohomology of D8

In A-infinity, Algebra, English, Homology and Homotopy, Mathematics.

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 [tex]H^*(D_8,\mathbb
F_2)[/tex].

^{i+1}, and the differentials looking like

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:

This describes the image [tex]f_1(x)[/tex] of our quasi-isomorphism [tex]\Gamma\to\Hom_\Lambda(P,P)[/tex].

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 [tex]f_1[/tex] 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
[tex](xy)[/tex], since [tex]f_1(x)f_1(x)f_1(y)f_1(y)=0[/tex], and we
define [tex]f_2(x^2,y)=f_1(x)f_2(x,y)[/tex] and
[tex]f_2(x,y^2)=f_2(x,y)f_1(y)[/tex].

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.

[tex]f_1(y)f_1(x)[/tex] is the same thing, but with ab and ba interchanged.

We want [tex]f_2(x,y)[/tex] and [tex]f_2(y,x)[/tex] 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 [tex]f_2(x,y)[/tex] has lower right entry a for all odd degree map components, and [tex]f_2(y,x)[/tex] has the entry above that b, same components. All even components vanish.

_{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,

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}.

_{3}(x,y,x) would be h with

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

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 set m_{4}(x,y,x,y)=m:sub:4(y,x,y,x)=z and
f_{4}=0, and stop calculating right here.