I am interested in $S^1$-actions on smooth, closed, and oriented manifolds $M$. I suppose that the action has a fixed point (I also suppose $M$ is connected). Let $\operatorname{Diff}(M)$ denote the diffeomorphism group of $M$. What are some examples of manifolds where the map

$$ H_1(S^1) \to H_1(\operatorname{Diff}(M))$$

is injective? I would be interested in getting as broad a class of examples as possible.



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