Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The fibre of the degree $3$ map, Anick spaces and the double suspension   Let $S^{2n+1}\{p\}$ denote the homotopy fibre of the degree $p$ self map of
$S^{2n+1}$. For primes $p \ge 5$, work of Selick shows that $S^{2n+1}\{p\}$
admits a nontrivial loop space decomposition if and only if $n=1$ or $p$.
Indecomposability in all but these dimensions was obtained by showing that a
nontrivial decomposition of $\Omega S^{2n+1}\{p\}$ implies the existence of a
$p$-primary Kervaire invariant one element of order $p$ in $\pi_{2n(p-1)-2}^S$.
We prove the converse of this last implication and observe that the homotopy
decomposition problem for $\Omega S^{2n+1}\{p\}$ is equivalent to the strong
$p$-primary Kervaire invariant problem for all odd primes. For $p=3$, we use
the $3$-primary Kervaire invariant element $\theta_3$ to give a new
decomposition of $\Omega S^{55}\{3\}$ analogous to Selick's decomposition of
$\Omega S^{2p+1}\{p\}$ and as an application prove two new cases of a
long-standing conjecture stating that the fibre of the double suspension
$S^{2n-1} \longrightarrow \Omega^2S^{2n+1}$ is homotopy equivalent to the
double loop space of Anick's space.

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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: mtrl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg