This paper considers the topological interference management problem in a partially connected K-user interference channel, where no channel state information at transmitters (CSIT) is available beyond the network topology knowledge. Due to the practical CSI assumption, this problem has recently received enough attention. In particular, it has been established that the topological interference management problem, in terms of degrees of freedom (DoF), is equivalent to the index coding problem with linear schemes. However, so far only a few index coding problems have been solved, and thus there is a lack of a systematic way to characterize optimal DoF of an arbitrary network topology. In this paper, we present a low-rank matrix completion (LRMC) approach to find linear solutions to maximize the achievable symmetric DoF for any given network topology. To decode the desired messages at each receiver, we also propose an LRMC based channel acquisition scheme, which can obtain interference-free measurements of the desired channel at each receiver while minimizing the pilot training length. To address the NP-hardness of the non-convex rank objective function in the resulting LRMC problem, we further present a Riemannian pursuit (RP) algorithm to solve it efficiently. This algorithm alternatively performs fixed-rank optimization using Riemannian optimization and rank increase by exploiting the manifold structure of the fixed-rank matrices. The LRMC approach aided by the RP algorithms not only recovers the existing optimal DoF results but also provides insights for general network topologies.