Let $p$ be a prime. Is $S^1$ a p-local CW-complex? Meaning, for any reduced homology theory $\overline{E}_*$, do we have $\overline{E}_*(S^1)=\overline{E}_*(S^1) \otimes_{\mathbb{Z}} \mathbb{Z}_{(p)}$?
Another question: If $\overline{H}_*(X, \mathbb{Z})=0$ then is $X$ $p$-local?
(Apologies if these questions indicate I am very confused about things).