On the Cohomology of Stable Two Stage Postnikov Systems
نویسندگان
چکیده
Introduction. Let £ = (£ , p, B, F) denote a two stage Postnikov system with stable fe-invariant. We announce results about H*(ÜE) as a Hopf algebra over the Steenrod algebra. Mod 2 cohomology is used exclusively. Unexplained notation is from [4] and [S]. 1 am grateful to D. Anderson, W. Massey, F . Peterson and H. Salomonsen for many useful remarks. We make the following assumptions on £, in addition to those of [5, p. 38]. F and B are simply connected products of finitely many Eilenberg-MacLane spaces. The nonzero homotopy groups of the factors of B are infinite cyclic or cyclic of order 2, k = 1, 2, • • • . All factors of F have Z% (cyclic group of order 2) as their only nonzero homotopy group. Results of [3], [4], [5] and [8] give H*(QE)^R®U(X'). The isomorphism is as algebras over Z2 and ® is over Z2. i?=i?(0£) = il*(£2£)/ker £2£* and X' = X'(ti%) is considered as known, [5, p. 54]. In general H*(&E) does not split this way as a Hopf algebra over Z2. The new result is Theorem A. I t gives iJ*(fi£) as a coalgebra over R. I t also gives information on the extension problem represented by the fundamental sequence of £2£ [5, p. 54]. This use of the Hopf algebra structure is well known, [ l ] , [5] and [ó].
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تاریخ انتشار 2007