Kameko's homomorphism and the algebraic transfer (Q309774)

The paper under review is a contribution to the study of the Steenrod algebra and related problems. The mod \(2\) Steenrod algebra \(\mathcal{A}_{2}\), an important tool in homotopy theory, can be defined with the help of the singular cohomology of the classifying space \(B\mathbb{Z}/2\). This cohomology is isomorphic to the polynomial ring on one variable of degree \(1\). Therefore, the Steenrod algebra acts upon the polynomial ring \(P_{k}\) on \(k\) variables, each of dimension \(1\), as it is isomorphic to the singular cohomology of the space \(B\left(\mathbb{Z}/2^{\oplus k}\right)\). The ring \(P_{k}\) also enjoys the action of the general linear groups \(\mathrm{GL}_{k}:=\mathrm{GL}_{k}\left(\mathbb{F}_{2}\right)\) induced by the action of the same group on \(\left(\mathbb{Z}/2\right)^{\oplus k}\). Having observed that this action commutes with the Steenrod one, Singer defined the morphism NEWLINE\[NEWLINE\varphi_{k}:\mathrm{Tor}_{k,k+n}^{\mathcal{A}_{2}}\left(\mathbb{F}_{2},\mathbb{F}_{2}\right)\to\left( \left(\mathbb{F}_{2}\otimes_{\mathcal{A}_{2}}P_{k}\right)^{\mathrm{GL}_{k}}\right)^{n}NEWLINE\]NEWLINE that bears his name [\textit{W. M. Singer}, Math. Z. 202, No. 4, 493--523 (1989; Zbl 0687.55014)]. Here, the upper index \(n\) denotes the subspace of homogeneous elements of degree \(n\). Note that the dual \(\varphi_{k}^{*}\) maps into the Ext-groups which form the \(E_{2}\)-term of the Adams spectral sequence for the stable homotopy groups of spheres. The study of \(\varphi_{k}\) has got quite a bit of attention and is the main goal of the article under review. In particular, the paper is concerned with the Singer conjecture, which predicts that the algebraic transfer \(\varphi_{k}\) is an epimorphism for all integers \(k\geq 1\). The verification of Singer himself when \(k=1\) and \(k=2\), that of Boardman when \(k=3\), and the study of the second author when \(k=4\) provide evidence for this conjecture. The main result of the article is an extension of the previous work to the case \(k=5\). The authors prove that if \(n=7\cdot 2^{s}-5\) for some integer \(s\geq 1\), then the group \(\left(\left(\mathbb{F}_{2}\otimes_{\mathcal{A}_{2}}P_{5}\right)^{\mathrm{GL}_{5}}\right)^{n}\) vanishes, resolving in the affirmative the Singer conjecture for such \(k\) and \(n\). These computations are carried out with the help of the Kameko homomorphism \(\widetilde{\mathrm{Sq}^{0}_{*}}\) which is the linear endomorphism of \(\left(\mathbb{F}_{2}\otimes_{\mathcal{A}_{2}}P_{k}\right)^{\mathrm{GL}_{k}}\) defined by \textit{M. Kameko} in his Ph.D. thesis [Products of projective spaces as Steenrod modules. Ann Arbor, MI: The Johns Hopkins University (1990)]. In particular, the authors show that there exist various cases in which the Kameko homomorphism is an isomorphism, extending a certain result achieved by \textit{N. H. V. Hung} [Trans. Am. Math. Soc. 357, No. 10, 4065--4089 (2005; Zbl 1074.55006)].NEWLINENEWLINE Editorial remark: The article contains results and sketches of proofs, the proofs will be published elsewhere.