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On mod \(p\) \(A_p\)-spaces. (English) Zbl 1373.55013

Let \(X\) be a \(p\)-local \(A_p\)-space for an odd prime \(p\) with cohomology ring \[ H^\ast(X, \mathbb{Z}/p\mathbb{Z})=\bigwedge (x_{2m_1-1}, \ldots, x_{2m_r-1}), \quad \text{deg}(x_{2m_i-1})=2m_i-1 \] where \(m_1\leq m_j\) for all \(j\). Then by Y. Hemmi [Hiroshima Math. J. 24, No. 3, 583–605 (1994; Zbl 0865.55008)] we have that there is a space \(R_p(X)\), called the modified projective space over \(X\), with a map \(\varepsilon : \Sigma X \to R_p(X)\); and in particular, in the present case its cohomology ring contains a polynomial algebra truncated at height \(p+1\) on generators \(y_{2m_1}, \ldots, y_{2m_r}\) satisfying the relations \(\varepsilon^\ast(y_{2m_i})=\sigma^\ast(x_{2m_i-1})\) and equipped with a mod 3 Steenrod algebra action. Using this result the authors prove that if we set \[ m=\text{gcd}\{m_i \mid m_i \leq pm_1 \}, \] then \(m \mid p-1\). They find that this fact is also applicable to the classification of rank 3, mod 3 \(A_3\)-spaces (or equivalently, rank 3 3-local homotopy associative \(H\)-spaces) \(X\). It can be actually used to detect the possible types of \(X\) and so in addition, by using the result for the 3-projective space \(P_3(X)\) over \(X\) by N. Iwase [Mem. Fac. Sci., Kyushu Univ., Ser. A 38, 285–297 (1984; Zbl 0555.55003)], the authors succeed in obtaining a list of such spaces \(X\).

MSC:

55P45 \(H\)-spaces and duals
55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
55N15 Topological \(K\)-theory
55P15 Classification of homotopy type
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