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The realizability of some finite-length modules over the Steenrod algebra by spaces. (English) Zbl 1446.55008

In this paper, the authors consider the cyclic module \(J:= \mathcal{A} / \mathcal{A} Sq^3\) over the mod \(2\) Steenrod algebra \(\mathcal{A}\), its dual \(J^\vee\), and the families of iterated doubles \(\Phi^i J\) and \(\Phi^i J^\vee\). These are important objects in homotopy theory: for instance, restricted to \(\mathcal{A} (1)\), \(J\) gives the exceptional element of the Picard group known as the Joker.
By [A. Baker, Homology Homotopy Appl. 20, No. 2, 341–360 (2018; Zbl 1401.55009)], these modules are realizable as the cohomology of a spectrum if and only if \(i \leq 2\). To refine unstably, the authors use the condition that a finite \(\mathcal{A}\)-module \(M\) is optimally realizable if there exists a space with reduced mod \(2\)-cohomology isomorphic to \(\Sigma^{\sigma (M)} M\), where \(\sigma(M)\in \mathbb{Z}\) is minimal such that this suspension is an unstable module.
The main result of the paper shows that \(\Phi^i J\) and \(\Phi^i J^\vee\) are optimally realizable for \(i\leq 2\); this completes the above picture elegantly. (The cases \(i<2\) were treated in [loc. cit.]; an alternative proof is given here.)
The authors give unified proofs of optimal realization for the family \(J\), \(\Phi J\), and \(\Phi^2 J\) by using appropriate skeleta of \(B\mathrm{SO}(3)\), \(B \mathrm{G}_2\), and \(B \mathrm{DW}_3\) (the exceptional \(2\)-compact group of W. G. Dwyer and C. W. Wilkerson [J. Am. Math. Soc. 6, No. 1, 37–64 (1993; Zbl 0769.55007)]) respectively and then killing cyclic \(\mathcal{A}\)-submodules as required.
For \(J\), it suffices to kill a top cohomology class of the \(6\)-skeleton of \(B\mathrm{SO}(3)\) by taking the \(6\)-skeleleton of the homotopy fibre of the map representing the class in \(H^6(B\mathrm{SO}(3); \mathbb{F}_2)\). For \(\Phi J\), mod-\(2\) singular cohomology is replaced by \(k\mathrm{O}\), real connective \(K\)-theory, and the authors give a geometric construction of the appropriate class in \(k\mathrm{O}^7 (BG_2)\).
For \(\Phi^2 J\), \(k\mathrm{O}\) is replaced by \(\mathrm{tmf}/2\), mod-\(2\) topological modular forms. Here the implementation relies upon constructing the appropriate non-trivial cohomology class in \((\mathrm{tmf}/2)^{14} (Y)\) for a space \(Y\) with specified cohomology. This is carried out by an Adams spectral sequence argument.
The authors suggest that this provides some evidence for existence of an appropriate real homotopy representation of \(\mathrm{DW}_3\), mirroring the case of \(\Phi J\).

MSC:

55P42 Stable homotopy theory, spectra
55S10 Steenrod algebra
55S20 Secondary and higher cohomology operations in algebraic topology
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