Let $\Omega^\mathrm{fr}_\ast \cong \pi_\ast S$ denote the graded ring of cobordism classes of framed manifolds, which, by the Pontryagin-Thom construction, is isomorphic (as a graded ring) to the stable homotopy groups of spheres. Hill, Hopkins, and Ravenel proved that the Kervaire invariant $\kappa:\Omega^\mathrm{fr}_{4k+2} \simeq \pi_{4k+2} S \to \mathbf{Z}/2$ is trivial unless $k = 0,1,3,7,15$; the final case, when $k=31$, remains open. One key reduction to stable homotopy theory is provided by a result of Browder's, which says that the Kervaire invariant vanishes except in dimensions of the form $2^{i+1}-2$, in which case it is detected by the element $h_i^2\in \mathrm{Ext}^{2,2^{i+1}}_{\mathcal{A}_\ast}(\mathbf{F}_2, \mathbf{F}_2) \cong \mathbf{F}_2$ in the $E_2$-page of the $2$-complete Adams spectral sequence.

One can ask an analogue of this question at odd primes. Define an element $b_i \in \mathrm{Ext}^{2,2p^i(p-1)}_{\mathcal{A}_\ast}(\mathbf{F}_p, \mathbf{F}_p)$ by the $p$-fold Massey product $-\langle h_i, \cdots, h_i \rangle$. Then, the question becomes: is the element $b_i$ a permanent cycle in the $p$-complete Adams spectral sequence? If $p\geq 5$, Ravenel proved that there are differentials $d_{2p-1}(b_i) = h_0 b_{i-1}^p$ for $i>1$ (so $b_i$ is not a permanent cycle for $i>1$); but $b_0$ is a permanent cycle (even at $p=3$) representing the element $-\langle \alpha_1, \cdots, \alpha_1 \rangle = \beta_1 \in \pi_{2p^2-2p-1} S$. At $p=3$, my understanding is that the question remains open.

My question is the following. Is there an analogue of Browder's theorem, which provides an interpretation in geometric topology (like the Kervaire invariant) of the fate of the elements $b_i$?



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