Generalised geometric weak conjecture on spherical classes and non-factorisation of Kervaire invariant one elements

TL;DR

This paper investigates the Curtis conjecture and spherical classes, demonstrating that certain elements do not factor through n-fold transfers for specific groups, and extends results related to the Kervaire invariant and weak conjectures.

Contribution

It proves that for n>2, the composition involving n-fold transfers is trivial or vanishes on elements of positive Adams filtration, and shows Kervaire elements do not factor through these transfers for certain groups.

Findings

The composition is trivial for n>2.

The image vanishes on elements with Adams filtration ≥1 for n=2.

Kervaire invariant one elements do not factor through n-fold transfers for n>1.

Abstract

This paper is on the Curtis conjecture. We show that the image of the Hurewicz homomorhism $h:\pi_*Q_0S^0\to H_*(Q_0S^0;\mathbb{Z})$, when restricted to product of positive dimensional elements, is determined by $\mathbb{Z}\{h(\eta^2),h(\nu^2),h(\sigma^2)\}$. Localised at $p=2$, this proves a geometric version of a result of Hung and Peterson for the Lannes-Zarati homomorphism. We apply this to show that, for $p=2$ and $G=O(1)$ or any prime $p$ and $G$ any compact Lie group with Lie algebra $\mathfrak{g}$ so that $\dim\mathfrak{g}>0$, the composition $${_p\pi_*}Q\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to {_p\pi_*}Q_0S^0\stackrel{h}{\to}H_*(Q_0S^0;\mathbb{Z}/p)$$ where $\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to S^0$ is the $n$-fold transfer, is trivial if $n>2$. Moreover, we show that for $n=2$, the image of the above composition vanishes on all elements of Adams filtration at least $1$, i.e. those elements of ${_2\pi_*^s}\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n}$ represented by a permanent cycle $\mathrm{Ext}_{A_p}^{s,t}(\widetilde{H}^*\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n},\mathbb{Z}/p)$ with $s>0$, map trivially under the above composition. The case of $n>2$ of the above observation proves and generalises a geometric variant of the weak conjecture on spherical classes due to Hung, later on verified by Hung and Nam. We also show that, for a compact Lie group $G$, Curtis conjecture holds if we restrict to the image of the $n$-fold transfer $\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to S^0$ with $n>1$. Finally, we show that the Kervaire invariant one elements $\theta_j\in{_2\pi_{2^{j+1}-2}^s}$ with $j>3$ do not factorise through the $n$-fold transfer $\Sigma^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to S^0$ with $n>1$ for $G=O(1)$ or any compact Lie group as above.