Towards a Browder theorem for spherical classes in $\Omega^lS^{n+l}$

TL;DR

This paper extends Browder's theorem to spherical classes in iterated loop spaces of spheres, establishing conditions under which classes are non-spherical and confirming Eccles' conjecture for certain finite loop spaces.

Contribution

It generalizes Browder's theorem to higher loop spaces and verifies Eccles' conjecture for finite loop spaces with loop count less than nine.

Findings

Classes with certain dimension properties are not spherical in $\

$ ext{l} ext{ in } ext{}\{4,5,6,7,8 ext{, only bottom cell or Hopf invariant one classes are spherical}

ext{Partial results on degenerate cases when } ext{dim} ext{ }\xi eq 2^t - 1, ext{ for } l > n

Abstract

According to Browder if $4n+2\neq 2^{t+1}-2$ then the Kervaire invariant of the cobordism class of a $(4n+2)$-dimensional manifold $M^{4n+2}$ vanishes and $M^{2^{t+1}-2}$ is of Kervaire invariant one if and only if $h_t^2\in\mathrm{Ext}(\mathbb{Z}/2,\mathbb{Z}/2)$ is a permanent cycle. On the other hand, according to Madsen if $4n+2\neq 2^t-2$ then $M^{4n+2}$ is cobordant to a sphere (hence of Kervaire invariant zero) and $M^{2^{t+1}-2}$ is not cobordant to a sphere (hence of Kervaire invariant one) if and only if certain element $p_{2^{t}-1}^2\in H_*QS^0$ is spherical. Moreover, it is known that $p_{2^t-1}^2$ is spherical if and only if $h_t^2$ is a permanent cycle in the Adams spectral sequence. Moreover, classes $p_{2n+1}^2\in H_*QS^0$ with $2n+1\neq 2^t-1$ are easily eliminated from being spherical. Hence, Browder's theorem admits a presentation and proof in terms of certain square classes being spherical in $H_*QS^0$ (see also work of Akhmetev and Eccles). In this note, we consider the problem of determining spherical classes $H_*\Omega^lS^{n+l}$ with $n>0$ and $4\leqslant l\leqslant +\infty$. We show (1) if $\xi^2\in H_*\Omega^lS^{n+l}$ is given with $\dim\xi+1\neq 2^t$ and $\dim \xi+1\equiv 2\textrm{ mod }4$ and $n>l$, then $\xi^2$ is not spherical. We refer to this as a generalised Browder theorem. We also present some partial results on the degenerate cases, corresponding to $\dim\xi\neq 2^t-1$, when $l>n$. (2) For $l\in\{4,5,6,7,8\}$ the only spherical classes in $H_*\Omega^lS^{n+l}$ arise from the inclusion of the bottom cell, or the Hopf invariant one elements. This verifies Eccles conjecture when restricted to finite loop spaces with $l<9$.