We use topological data analysis (TDA) to study how data transforms as it passes through successive layers of a deep neural network (DNN). We compute the persistent homology of the activation data for each layer of the network and summarize this information using persistence landscapes. The resulting feature map provides both an informative visualization of the network and a kernel for statistical analysis and machine learning. A statistical test shows that it correlates with classification accuracy. We observe that the topological complexity often increases with training and that the topological complexity does not decrease with each layer.