Population displacement is a housing-related involuntary residential dislocation. It has become increasingly widespread in many cities, particularly in neighbourhoods undergoing rapid economic and demographic change, and measuring it is essential to assess the social consequences of urban transformation and housing market pressures. Despite its relevance, quantifying displacement presents difficulties due to limited replicability across cities and time periods and the need to analyse long time spans: displacement is a gradual process, impossible to capture in one data snapshot. We introduce a novel tool to overcome these difficulties. Using publicly available address change data, we construct four cubical complexes simultaneously incorporating geographical and temporal information of people moving, and analyse using Topological Data Analysis tools. Finally, we demonstrate this method through a 20-year case study in Madrid, Spain. The results reveal its ability to capture displacement and identify the neighbourhoods and years affected--patterns not observable from raw address change data.