Human motion analysis is a very important research topic in the field of computer vision, as evidenced by a wide range of applications such as video surveillance, medical assistance and virtual reality. Human motion analysis concerns the detection, tracking and recognition of human activities and behaviours. The development of low-cost range sensors enables the precise 3D tracking of body position. The aim of this paper is to present and evaluate a novel method based on topological data analysis (TDA) for motion capture (kinematic) processing and human action recognition. In contrast to existing methods of this type, we characterise human actions in terms of topological features. The recognition process is based on topological persistence which is stable to perturbations. The advantages of TDA are noise resistance and the ability to extract global structure from local information. The method we proposed in this paper deals very effectively with the task of human action recognition, even on the difficult classes of motion found in karate techniques. In order to evaluate our solution, we have performed three-fold cross-validation on a data set containing 360 recordings across twelve motion classes. The classification process does not require the use of machine learning and dynamical systems theory. The proposed classifier achieves a total recognition rate of 0.975 and outperforms the state-of-the-art methods (Hachaj, 2019) that use support vector machines and principal component analysis-based feature generation.