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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

##### MSC:
 55S10 Steenrod algebra 55P25 Spanier-Whitehead duality 55P99 Homotopy theory 55P42 Stable homotopy theory, spectra
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