This paper proposes a novel framework for activity recognition from 3D motion capture data using topological data analysis (TDA). We extract point clouds describing the oscillatory patterns of body joints from the principal components of their time series using Taken's delay embedding. Topological persistence from TDA is exploited to extract topological invariants of the constructed point clouds. We propose a feature extraction method from persistence diagrams in order to generate robust low dimensional features used for classification of different activities. Our experimental results demonstrate high separability of generated features, and as a result a high accuracy of classification.