A polymer containing a carboxyl group, a preparation method and an application thereof, a supported catalyst and a preparation method thereof and preparation methods of a penem antibiotic intermediate are disclosed. The polymer is prepared by polymerizing three monomers with different structures. The carboxyl group-containing polymer is a cross-linked polymer, and a polymer chain contains a large number of benzene rings, which can improve the rigidity and hardness of the polymer, thus effectively improving the mechanical properties of the polymer. Meanwhile, in the polymer, the carboxyl group is used as a main functional group, and is used as a carrier to prepare, by means of a coordination reaction between the carboxyl group and a heavy metal, a supported metal catalyst which has better connection stability between the metal and the polymer. The above two factors can improve the stability of the supported metal catalyst, such that the catalyst can be recycled without losing the catalytic activity. Meanwhile, loss of a heavy metal active ingredient and production cost can be reduced.