| Let A (n) be the subalgebra of the mod-2 Steenrod algebra generated by {1, Sq1,…, Sq 2n}. I hope to construct a 2-group G with the property H*(G, Z /2) ≅ ExtA2 ( Z /2, Z /2). The difficulty in this construction is due to the different actions of Sq0 on the cohomology of a group and the cohomology of a Hopf algebra.;I use the May spectral sequence to calculate ExtA2 ( Z /2, Z /2). I then use central extensions of groups corresponding to some of the May spectral sequence differentials. I use the Lyndon-Hochschild-Serre (LHS) spectral sequence to calculate the group cohomology for each central extension, and also maps to the LHS spectral sequences for certain subgroups. I use additional central group extensions for each time the Sq 0 action on the May spectral sequence differs from the Sq 0 action on the LHS spectral sequence.;This process works for realizing some of the May spectral sequence differentials in the cohomology of groups, but not all. In particular, there are two differentials in the calculation of ExtA2 ( Z /2, Z /2) that cannot be constructed in this way. Thus, the result of Sq0 acting on ExtA2 ( Z /2, Z /2) cannot necessarily be accomplished using extra central extensions of groups. |
