The cerebrospinal fluid (CSF) of 19 subjects who received a clinical diagnosis of Alzheimer's disease (AD) as well as of 5 pathological controls have been collected and analysed by Raman spectroscopy (RS). We investigated whether the raw and preprocessed Raman spectra could be used to distinguish AD from controls. First, we applied standard Machine Learning (ML) methods obtaining unsatisfactory results. Then, we applied ML to a set of topological descriptors extracted from raw spectra, achieving a very good classification accuracy (>87%). Although our results are preliminary, they indicate that RS and topological analysis together may provide an effective combination to confirm or disprove a clinical diagnosis of AD. The next steps will include enlarging the dataset of CSF samples to validate the proposed method better and, possibly, to understand if topological data analysis could support the characterization of AD subtypes.