I was wondering if one could provide intuition to the following result. Assume $X$ and $Y$ are path connected.
If $\pi_n(X)\cong \pi_n(Y)$ then for all $n$, then $H^*(X)\cong H^*(Y)$.
This sounds like a general hurewicz type argument but I dont know
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https://math.stackexchange.com/questions/99302/spaces-with-equal-homotopy-groups-but-different-homology-groups,https://math.stackexchange.com/questions/426212/stronger-two-space-formulation-of-hurewicz-theorem-about-homotopy-and-homology-g?rq=1 – Moishe Kohan May 21 '23 at 19:03
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@MoisheKohan Thanks! – May 21 '23 at 19:05
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1Does this answer your question? Stronger two-space formulation of Hurewicz theorem about homotopy and homology groups – Moishe Kohan May 21 '23 at 19:05
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@MoisheKohan Thanks yes it does! I asked a new question of what I had in mind. – May 21 '23 at 19:07