This paper presents a novel topology-driven trajectory optimization strategy for ground robots navigating in complex, dynamic environments. The proposed method, called Topology-driven Model Predictive Control (T-MPC), aims to enhance the robot's behavior and efficiency by planning multiple distinct evasive trajectories. The approach involves a global planner that generates trajectories in distinct homotopy classes, which are then optimized by local planners working in parallel. Each local planner is constrained to a specific homotopy class, ensuring that the robot executes a feasible trajectory with the lowest cost. The paper demonstrates the effectiveness of T-MPC through simulations and real-world experiments, showing that it leads to faster and safer trajectories compared to existing planners. The key contributions include a planning framework that optimizes trajectories in multiple distinct homotopy classes in parallel, a fast guidance planner that computes homotopy distinct trajectories, and a robust local planner that enforces the execution of trajectories in distinct homotopy classes. The method is validated on a mobile robot navigating among pedestrians, demonstrating its ability to handle dynamic environments and improve upon existing planners in terms of task duration, safety, and runtime.This paper presents a novel topology-driven trajectory optimization strategy for ground robots navigating in complex, dynamic environments. The proposed method, called Topology-driven Model Predictive Control (T-MPC), aims to enhance the robot's behavior and efficiency by planning multiple distinct evasive trajectories. The approach involves a global planner that generates trajectories in distinct homotopy classes, which are then optimized by local planners working in parallel. Each local planner is constrained to a specific homotopy class, ensuring that the robot executes a feasible trajectory with the lowest cost. The paper demonstrates the effectiveness of T-MPC through simulations and real-world experiments, showing that it leads to faster and safer trajectories compared to existing planners. The key contributions include a planning framework that optimizes trajectories in multiple distinct homotopy classes in parallel, a fast guidance planner that computes homotopy distinct trajectories, and a robust local planner that enforces the execution of trajectories in distinct homotopy classes. The method is validated on a mobile robot navigating among pedestrians, demonstrating its ability to handle dynamic environments and improve upon existing planners in terms of task duration, safety, and runtime.