The scarcity of electric vehicle (EV) charging stations is a critical barrier to the widespread adoption of EVs. Manufacturers face uncertainty in participating in building charging stations and developing station types. We investigate the manufacturer's optimal building strategy by developing a game-theoretic model. The results show that, when the construction cost is relatively low, the manufacturer has an incentive to build charging stations; otherwise, the manufacturer should make a trade-off between competition and cooperation with the platform. When the construction cost of super charging stations goes high and the network effect is not as large, the manufacturer decides to build general charging stations. Regardless of the type of charging stations, the scale of charging stations and EV demand always increases, but the platform's construction scale does not always enlarge. Interestingly, if the charging competition hurts the platform's profit, the manufacturer lowers EV price. The EV-to-charging station ratio can only decrease and the charging service level can only improve if the increment in EV demand is lower than that in charging stations; otherwise, the charging service level may worsen. Moreover, the change in both the fraction of consumers and construction cost can achieve win-win or win-lose situations for both manufacturers and platforms. Finally, although there can be a win-win situation for both entities in the joint mode, the self-built mode is always the dominant strategy. These insights not only contribute to guiding the manufacturer's strategies for building charging stations to promote EV adoption, but also provide advice to improve EV charging service level.