I have previously calculated the A:sub:`∞-structure for the cohomology ring of D8 <http://blog.mikael.johanssons.org/archive/2006/11/an-a-structure-on-the-cohomology-of-d8/>`__. Now, while trying to figure out how to make my work continue from here, I tried working out what algebra this would have come from, assuming that I can adapt Keller's higher multiplication theorem to group algebras.
A success here would be very good news indeed, since for one it would indicate that such an adaptation should be possible, and for another it would possibly give me a way to lend strength both to the previous calculation and to a conjecture I have in the calculation of group cohomology with A∞ means.
So, we start. We recover, from the previous post, the structure of the cohomology ring as k[x,y,z]/(xy), with x,y in degree 1, and z in degree 2. Furthermore, we have a higher operation, m:sub:`4`, with m:sub:`4`(x,y,x,y)=m:sub:`4`(y,x,y,x)=z.
As such, this is an unqualified success. We recover our original group algebra presentation from the A∞-structure, and thus should be able to do similarily as a test for completion of future calculations as well. This, of course, needs to be proven before relied upon, but it lends credence to my hopes.