A product decomposition of \(\Omega^3_0\Sigma\mathbf RP^2\)
Let \(P^n(2)\) be the \(n\)-dimensional mod 2 Moore space. The author proves that there is a homotopy equivalence \[ \Omega^3_0P^3(2)\simeq \Omega^2S^3\langle 3\rangle\times \Omega^3_0(P^6(2)\vee \Sigma(\mathbb{R} P^4/\mathbb{R} P^1)) \] localized at 2. \((\Omega^3_0 X\) denotes the base-point path-connected component of the triple loop space of the space \(X\)).
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