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On topologically realizing modules over the Steenrod algebra. (English) Zbl 0849.55022

The \(\mod p\) cohomology of a space or a spectrum has the additional structure as a module over the Steenrod algebra \(A_p\). For a space, this structure is unstable in the sense (for \(p = 2\)) \(S^i_q x = 0\) when \(i > \text{deg } x\). For modules over \(A_p\), unstable or not there is a multitude of questions about realizing these modules as the cohomology of a space or spectrum. The problem of maps of Hopf invariant one [J. F. Adams, ibid. 72, 20-104 (1960; Zbl 0096.17404)] is easily seen to be of this type.
The author is interested in the Realization Conjecture which says that if \(H^*(X; Z/p)\) is a finitely generated \(A_p\)-module, then it is actually finite. Here \(X\) is a space or a spectrum with unstable cohomology. The stable analogue is false as \(A_p\) can be realized (one generator). This paper consists of a deep analysis of conditions for or related to the realization conjecture. For example, the author shows that the conjecture is true when \(H^* (X; Z/p)\) is annihilated by the Bockstein in high degrees.
The methods are sophisticated including the \(T\)-technology [J. Lannes, Sur les éspaces fonctionnels dont la source est le classifiant d’un \(p\)-groupe abelien élémentaire, Publ. Math., Inst. Hautes Étud. Sci. 75, 135-244 (1992)], the work of L. Schwartz [“Unstable modules over the Steenrod algebra and Sullivan’s fixed point conjecture”, Univ. Chicago Press, 1994], and the work of P. Krasoń and the author [J. Algebra 163, 281-294 (1994; Zbl 0832.20063)]. Details are rather too complex to include here.

MSC:

55S10 Steenrod algebra
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