  Many-body effects on superfluidity and transition temperatures are calculated
for optical lattices and uniform systems with ultracold multi-component Fermi
gases. The induced interactions depend sensitively on the interactions between
the multi-components and their densities. The s- and d-wave pairing gaps and
critical temperatures are calculated for optical lattices at low (dilute)
filling as well as near half filling to leading orders in the interaction
strength with and without induced interactions. They can deviate strongly from
dilute two-component systems and affect the phase diagram. In two dimensional
optical lattices the induced interactions are singular at half filling and
strongly affect s- and d-wave superfluidity.
