In mobile edge computing (MEC) systems, the wireless channel condition is a critical factor affecting both the communication power consumption and computation rate of the offloading tasks. This paper exploits the idea of cooperative transmission and employing reconfigurable intelligent surface (RIS) in MEC to improve the channel condition and maximize computation efficiency (CE). The resulting problem couples various wireless resources in both uplink and downlink, which calls for the joint design of the user association, receive/downlink beamforming vectors, transmit power of users, task partition strategies for local computing and offloading, and uplink/downlink phase shifts at the RIS. To tackle the challenges brought by the combinatorial optimization problem, the group sparsity structure of the beamforming vectors determined by user association is exploited. Furthermore, while the CE does not explicitly depend on the downlink phase shifts, instead of simply finding a feasible solution, we exploit the hidden relationship between them and convert this relationship into an explicit form for optimization. Then the resulting problem is solved via the alternating maximization framework, and the nonconvexity of each subproblem is handled individually. Simulation results show that cooperative transmission and RIS deployment can significantly improve the CE and demonstrate the importance of optimizing the downlink phase shifts with an explicit form.