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Homotopy classification of spaces with interesting cohomology and a conjecture of Cooke. I. (English) Zbl 0843.55007

The purpose of this paper is to obtain a classification up to \(p\)-completion of all homotopy types with \(\text{mod } p\) cohomology of the form \(B_{i,r}\) and \(A_r\) for \(i \geq 0\), \(r\) dividing \(p - 1\), and \(p\) denotes an odd prime. As graded algebras (subscripts denote degrees), \(B_{i,r} \cong {\mathbf F}_p[x_{2p^i r}] \otimes E(y_{2p^i r + 1})\) and the Steenrod algebra action is determined by \(\beta(x) = y\) and \(P^{p^i} (y) = (r - 1)x^s y\). The algebra \(A^{(k)}_r\), \(k \geq 0\), is isomorphic to \(B_{0,r}\) but the action of the Steenrod algebra is different for \(A^{(k)}_r\) the relation \(P^1(y) = rx^s y\) holds and \(\beta_{(k+1)} x = y\) where \(\beta_{(k+1)}\) denotes the Bockstein homomorphism of order \(k + 1\).
First the authors prove that if \(H^* (X, {\mathbf F}_p) \cong B_{i,r}\) then \(i \leq 1\), and then prove the Cooke conjecture in full generality. Secondly they obtain a complete list of \(p\) complete homotopy types realizing \(A^{(k)}_r\), \(B_{0,r}\) and \(B_{1,r}\). Finally they study suspensions of all the spaces constructed. It turns out that \(\ell\)-fold suspensions are not homotopy equivalent.

MSC:

55P15 Classification of homotopy type
55S10 Steenrod algebra
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