Let $H\mathbb{Z}$ be the Eilenberg Maclane spectrum by $(H\mathbb{Z})^k=\tilde{\mathbb{Z}}[S^k]$. Here $S^1 = \Delta[1]/\partial \Delta[1]$ is the simplicial circle and $S^k = S^1\wedge \cdots \wedge S^1$ ($k$ times), and $(\tilde{\mathbb{Z}}[S^k])_n=\mathbb{Z}(S^k)_n/\mathbb{Z}*$, where the $*$ is the basepoint.

It is known that (as in Motivic Homotopy Theory, 6.4.1, p23) $$\tilde{H}_n(X) = \pi_n(H\mathbb{Z} \wedge X)$$ where the homotopy is the stable homotopy. The proof may be given by $$\tilde{H}_n(X) = \tilde{H}_{n+k}(S^k\wedge X) = \pi_{n+k}(\tilde{\mathbb{Z}}[S^k\wedge X])$$ and $$\pi_{n+k}((H\mathbb{Z}\wedge X)^k) = \pi_{n+k}(\tilde{Z}[S^k]\wedge X)\rightarrow \pi_{n+k}(\tilde{Z}[S^k\wedge X])$$ with the last map being an isomorphism for $k \ge n$.

What I do not know is, why the last map which is induced by $\tilde{Z}[S^k]\wedge X\rightarrow \tilde{Z}[S^k\wedge X]$ is a weak equivalence.


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