On 3-manifolds with locally standard -actions

We investigate what kind of closed 3-manifolds can admit locally standard ( Z 2 ) 3 -actions. In particular, we will prove that for a closed connected 3-manifold M with H 1 ( M ; Z 2 ) = 0 , M admits a locally standard ( Z 2 ) 3 -action if and only if M is a connected sum of 8 copies of a Z 2 -homology sphere N. So if such an M is irreducible, it must be homeomorphic to S 3 . Moreover, the argument can be extended to study orientable rational homology 3-spheres M with H 1 ( M ; Z 2 ) ≅ Z 2 which admits locally standard ( Z 2 ) 3 -actions.