Perfect, locally free groups exist. Such a thing has vanishing $H_1(G,\mathbb Z)$, has $H_p(G,M)=0$ for all $p\geq2$ and all $M$, and is not free.
A. J. Berrick constructs an explicit example here.
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Mariano Suárez-Álvarez
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Mariano Suárez-Álvarez
- 47.3k
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- 147
- 264