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