Are there any examples of compact connected Lie groups with vanishing first homology groups in dimension $3$ different from $S^3$?
2026-03-29 05:11:39.1774761099
Acyclic compact Lie groups of dimension 3
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$S^3$ isn't acyclic, if by acyclic you mean having vanishing homology in positive degree; in fact no compact Lie group of positive dimension is acyclic, because they're closed and orientable and hence have nontrivial top homology.
The complete list of $3$-dimensional compact connected Lie groups is
There are lots more disconnected ones, e.g. any of these times a finite group. Of these, $SU(2)$ is the only one that's simply connected. Is that what you meant?