Is it possible to get a space (may not be a CW complex) which has some non zero homotopy group, but all of whose stable homotopy groups are zero?
2026-03-26 16:26:35.1774542395
Stable homotopy type of a space
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I'll turn my comment into an answer: David White's answer to a question on mathoverflow provides a space $X$ which is not contractible, but its suspension is. In particular, $\pi_1(X) \not \cong 0$, but all of the homotopy groups of $\Sigma X$ are zero, and hence the stable homotopy groups of $X$ are zero.