zbMATH — the first resource for mathematics

Finite complexes with A(n)-free cohomology. (English) Zbl 0568.55021
Let p be a prime and let A be the mod p Steenrod algebra. For \(n\geq 0\) let A(n) be the subalgebra of A generated by \(\beta\), \(P^ 1,...,P^{p^{n-1}}\) and let P(n) be the subalgebra of A generated by \(P^ 1,...,P^{p^ n}\). (If \(p=2\) interpret \(P^ i\) as \(Sq^{2i}\) and take P(n) as a subalgebra of A/(\(\beta)\).) These subalgebras are finite dimensional, and it is a central problem in homotopy theory to determine which finite dimensional subalgebras of A can be realized as the cohomology of a finite CW complex. The author uses invariant theory to prove that the algebra structures on the A(n) and P(n) extend to self- dual A-module structures. He then constructs finite CW complexes \(X_ n\), \(n\geq 0\), whose mod p cohomology is free over A(n-1) and hence over \(E(Q_ 0,...,Q_{n-1})\). These \(X_ n\) are also Spanier-Whitehead self-dual.
Reviewer: S.O.Kochman

55S10 Steenrod algebra
55P25 Spanier-Whitehead duality
55P99 Homotopy theory
55P42 Stable homotopy theory, spectra
Full Text: DOI