The algebraic transfer for the real projective space (Q666720)

For \(M\) a module over the mod \(2\) Steenrod algebra \(\mathcal{A}\), the \(s\)-fold algebraic transfer [\textit{W. M. Singer}, Math. Z. 202, No. 4, 493--523 (1989; Zbl 0687.55014)] \[ \mathrm{Hom}_{\mathcal{A}} (H^* V_s \otimes M, \mathbb{F}_2) _{GL_s} \rightarrow \mathrm{Ext}^s_{\mathcal{A}} ( \Sigma^{-s} M, \mathbb{F}_2) \] is induced by the class \(e_s \in \mathrm{Ext}^s_{\mathcal{A}} ( H^* V_s ,\Sigma^{-s} \mathbb{F}_2)\) that represents the \(s\)-iterated \(\mathbb{Z}/2\)-transfer, where \(H^* V_s\) is the cohomology of the rank \(s\) elementary abelian \(2\)-group \(V_s\). The authors announce a panoply of results for this transfer, concentrating on the case \(M=\tilde{H}^* (\mathbb{R}P^\infty)\), which is of especial interest due to the relationship with the case \(M = \mathbb{F}_2\). For instance, they state a multiplicative property of the algebraic transfer, leading to examples showing that it is non-trivial but not surjective. To analyse the kernel, they exploit the compatibility between Kameko's squaring operation \(Sq^0\) (a fundamental tool in studying \(H^*V_s \otimes_{\mathcal{A}} \mathbb{F}_2\)) and the operation \(Sq^0\) on \(\mathrm{Ext}^*_{\mathcal{A}} ( \tilde{H}^* (\mathbb{R}P^\infty), \mathbb{F}_2)\). Developing results and methods of [\textit{N. H. V. Hung}, Trans. Am. Math. Soc. 357, No. 10, 4065--4089 (2005; Zbl 1074.55006)], they deduce that the \(s\)-fold transfer has non-trivial kernel in infinitely many degrees if \(s>0\). For this they introduce \textit{critical elements}, defined in terms of the \(Sq^0\) actions and the transfer.