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For a based space $X$, let $Q(X) = \Omega^\infty \Sigma^\infty (X)$. Let $D_2(X) = (X\wedge X) \wedge_{\Bbb Z_2} E\Bbb Z_2$ denote the quadratic construction. Then one has a pair of maps $$ X \overset{E}\longrightarrow Q(X) \overset{H}\longrightarrow Q(D_2(X)) $$ where $E$ is adjoint to the identity map and $H$ is the stable Hopf invariant (often attributed to Segal and Snaith).

What is the earliest reference in the literature which says in effect that the above is homotopy fiber sequence in the metastable range (roughly three times the connectivity of $X$)?

This was presumably known in the 1970s, but I could not find the explicit statement.

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I think

Theorem 1.11. There is a space $\Gamma_L(X) = S^{L-1} \ltimes_T (X \wedge X)$, which is $(2n-1)$-connected whenever $X$ is $(n-1)$-connected, and an exact sequence valid in the metastable range ($i < 3n-2$) $$ \dots \overset{E}\to \pi_{i+L}(\Sigma^L X) \overset{H}\to \pi_i(\Gamma_L(X)) \overset{\partial}\to \pi_{i-1}(X) \overset{E}\to \pi_{i+L-1}(\Sigma^L X) \to \dots \,. $$

in

Milgram, R. James
Unstable homotopy from the stable point of view.
Lecture Notes in Mathematics, Vol. 368
Springer-Verlag, Berlin-New York, 1974. iv+109 pp.

is such a statement for $L$ sufficiently large. (I have inserted a comma.)

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