\(\mathcal Q\)-subalgebras, Milnor basis, and cohomology of Eilenberg-MacLane spaces
The \(\text{mod} p\) cohomology of Eilenberg-MacLane spaces was determined in the early 1950's by J. P. Serre for \(p=2\) and by H. Cartan for \(p\) odd. The description used the Steenrod algebra bases consisting of admissible monomials of Steenrod squares in case \(p=2\) and of Steenrod reduced powers and Bockstein operators for \(p\) odd. However, for many purposes the Milnor bases [\textit{J. W. Milnor}, Ann. Math., II. Ser. 67, 150-171 (1958; Zbl 0080.38003)] is more convenient. In the paper the author describes the \(\text{mod} p\) cohomology of various Eilenberg-MacLane spaces in terms of the Milnor bases. In particular the following cases are examined: (1) \(\text{mod} p\) cohomology of \(\text{mod} p^n\) Eilenberg-MacLane spaces at an odd prime \(p\); (2) \(\text{mod} p\) cohomology of integral Eilenberg-MacLane spaces at an odd prime \(p\); (3) \(\text{mod} 2\) cohomology of \(\text{mod} 2^n\) Eilenberg-MacLane spaces at prime 2; (4) \(\text{mod} 2\) cohomology of integral Eilenberg-MacLane spaces at prime 2. The author gives the formulae for excess of elements in the Milnor basis. Certain natural, polynomial subalgebras of the cohomology algebras of Eilenberg-MacLane spaces are also introduced and investigated.
scientific article; zbMATH DE number 1308327
scientific article