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The Lannes-Zarati homomorphism and decomposable elements. (English) Zbl 1422.55035
Let \(H:{\pi_n}Q_0X\to H_nQ_0X\) denote the Hurewicz homomorphism, where \(X\) is a pointed \(CW\)-complex (in practice of finite type), \(Q_0X\) is the base point component of \(QX=\mathrm{colim}\ \Omega^i\Sigma^iX\), \({\pi_*}\) stands for the \(2\)-primary part of homotopy and \(H_*\) denotes the \(\mathbb{Z}/2\)-homology.
The problem of determining the image of \(H\), with a special interest in the case of \(X=S^0\), is a long standing problem in stable homotopy theory going back to work of E. B. Curtis [Ill. J. Math. 19, 231–246 (1975; Zbl 0311.55007)]. The image in the case of \(X=S^0\) is predicted by a conjecture of Curtis which reads as follows.
Curtis conjecture. [loc. cit., Theorem 7.1] Let \(n>0\). For \(X=S^0\), only Hopf invariant one and Kervaire invariant one elements map nontrivially under \(H\).
The Hopf invariant one elements are detected by Steenrod operations \(Sq^{2^j}\) in their mapping cone, and the Kervaire invariant one elements are detected by the Adams operation associated to the Adem relation \(Sq^{2^j}Sq^{2^j}=\sum_{k=0}^{j-1} Sq^{2^{j+1}-2^k}Sq^{2^k}\). This motivates the following geometric conjecture of Eccles when \(X\) is assumed to a CW-complex as above.
Eccles conjecture. [H. Zare, Q. J. Math. 70, No. 3, 859–878 (2019; Zbl 1460.55014), Conjecture 1.2] Suppose \(X\) is path connected. If \(H(\alpha)\neq 0\) then the stable adjoint of \(\alpha\), say \(\widetilde{\alpha}:S^n\not\to X\) is either detected by a primary operation in its stable mapping cone or \((\widetilde{\alpha})_*\neq 0\).
This latter conjecture is more of a geometric nature, and it is known that Eccles’ conjecture for \(X=P\), the infinite dimensional projective space, implies Curtis’ conjecture [loc. cit., Lemma A.1, Lemma A.2]. On the other hand, since \(\pi_nQ_0X\simeq\pi_n^sX\) the \(n\)-th stable homotopy of \(X\) which is computed using the Adams spectral sequence techniques, it is desirable to have an algebraic version of either of above conjectures. This is done by studying a graded associated version of \(H\) which is known as the Lannes-Zarati-homomorphism; recently N. J. Kuhn has studied this homomorphism from a geometric point of view [Invent. Math. 214, No. 2, 957–998 (2018; Zbl 1403.55007)]. The \(s\)-th Lannes-Zarati homomorphism has the form \[ \varphi^M_s:\mathrm{Ext}_A^s(M,\mathbb{F}_2)\to (\mathbb{F}_2\otimes_A R_sM)^* \] where \(M\) is an unstable \(A\)-module with \(A\) being the mod \(2\) Steenrod algebra. The target of this homomorphism when \(M=\widetilde{H}^*X\) computes \(\pi_*^sX\). Note that Hopf invariant one elements are detected by permanent cycles living in the \(1\)-line of the ASS for \(\pi_*^s\), and the Kervaire invariant one elements are detected by permanent cycles living in the \(2\)-line of the ASS. It is also known that the other elements of \(\pi_*^s\) detected by permanent cycles living in the \(2\)-line map trivial under \(H\). Moreover, note that through the Kahn-Priddy map \(P\not\to S^0\) these elements pull back to elements of \(\pi_*^sP\) which are of Adams filtration one less than that in \(\pi_*^s\). Now, in this language a conjecture, similar to the Eccles conjecture but not necessarily equivalent to it, which is due to Hu’ng, reads as follows.
Hu’ng conjecture. Suppose \(X\) is path connected, and \(\alpha\in \pi_*^sX\) is of Adams filtration at least \(3\). Then \(H(\alpha)=0\).
Finally, we record the algebraic version of this latter conjecture when \(M\) is as stated in the paper.
The generalised algebraic spherical class conjecture. Suppose \(i>2\) and \(s>0\). Then, the Lannes-Zarati homomorphism \[ \varphi^M_s:\mathrm{Ext}_A^{s,s+i}(M,\mathbb{F}_2)\to (\mathbb{F}_2\otimes_A R_sM)^*_i \] vanishes for any unstable module \(M\).
Let’s note that originally the conjecture is stated for \(M=\widetilde{H}^*B\mathbb{F}_2^{\times k}\) [N. H. V. Hu’ng, Trans. Am. Math. Soc. 349, No. 10, 3893–3910 (1997; Zbl 0902.55004), Conjecture 1.2] (also compare to the related conjecture [loc. cit., Conjecture 1.3] known as the weak algebraic conjecture on spherical classes) and it seems to the reviewer that with an eye on the possible applications to Curtis’ conjecture assuming that \(M\) is reduced cohomology of a path connected space is necessary. An important elimination of the image of \(\varphi_s^M\) is due to N. H. V. Hu’ng and F. P. Peterson [Math. Proc. Camb. Philos. Soc. 124, No. 2, 253–264 (1998; Zbl 0906.55013), Theorem 5.4] which shows that for \(s>0\), in positive stems, and \(M=\mathbb{F}=H^*S^0\), \(\varphi_s^M=0\) on decomposable elements.
The paper under review generalies the result of Hu’ng and Peterson in the following way. At least in favourable cases when \(M=\widetilde{H}^*X\) then \(\mathrm{Ext}_A(M,\mathbb{F}_2)\) is an \(\mathrm{Ext}_A(\mathbb{F}_2,\mathbb{F}_2)\)-module. In this sense, one can talk about decomposable elements in \(\mathrm{Ext}(M,\mathbb{F}_2)\) which are of the form \(\alpha\beta\) with \(\alpha\in \mathrm{Ext}_A^{p,q}(\mathbb{F}_2,\mathbb{F}_2)\) and \(\beta\in\mathrm{Ext}_A^{s,t}(M,\mathbb{F}_2)\). The main result of this paper, recorded as Theorem 1.3, states that \(\varphi^M_s\) vanishes on certain decomposable elements \(\alpha\beta\in \mathrm{Ext}(M,\mathbb{F}_2)\) assuming certain grading conditions. As an application, the author reproves the result of Hu’ng and Peterson which is recorded as Theorem 1.4. The paper is finished by proving a vanishing result for \(\varphi^{\widetilde{H}^*P}_5\). The method of proof is algebraic, by looking at the relations and the interactions between various algebras such as the Dickson algebra, Mui algebra, Steenrod algebra, as well as detailed computations.
Reviewer: Hadi Zare (Tehran)
55T15 Adams spectral sequences
55S10 Steenrod algebra
Full Text: DOI
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