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On the metastable homotopy of $$\bmod 2$$ Moore spaces. (English) Zbl 1354.55005
Let $$P^n(2)=\Sigma^{n-2}\mathbb{R}\mathrm{P}^2$$ be the $$n$$ dimensional mod $$2$$ Moore space with $$n\geq 3$$. It is well-known that the metastable homotopy of $$P^n(2)$$ has an exponent dividing $$8$$ and this leads to the natural question whether the metastable homotopy of $$P^n(2)$$ has an exponent $$4$$. It is also known that it has an exponent $$8$$ when $$n\equiv 2, 3$$ mod $$4$$.
In this paper, the authors consider this question for the case $$n\equiv 0$$ mod $$4$$, and they prove that the homotopy group of the double loop space $$\Omega^2P^{4n}(2)$$ has the multiplicative exponent $$4$$ below the range of $$4$$ times connectivity by using the Cohen group for displaying the explicit obstructions to the $$4$$th power map on $$\Omega P^n(2)$$ and shuffle relations with Hopf invariants on general configuration spaces.

##### MSC:
 55Q52 Homotopy groups of special spaces 55Q05 Homotopy groups, general; sets of homotopy classes 55P35 Loop spaces 14F35 Homotopy theory and fundamental groups in algebraic geometry 55Q20 Homotopy groups of wedges, joins, and simple spaces
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