In recent years, cosmic shear has emerged as a powerful tool for studying the statistical distribution of matter in our Universe. Apart from the standard two-point correlation functions, several alternative methods such as peak count statistics offer competitive results. Here we show that persistent homology, a tool from topological data analysis, can extract more cosmological information than previous methods from the same data set. For this, we use persistent Betti numbers to efficiently summarise the full topological structure of weak lensing aperture mass maps. This method can be seen as an extension of the peak count statistics, in which we additionally capture information about the environment surrounding the maxima. We first demonstrate the performance in a mock analysis of the KiDS+VIKING-450 data: We extract the Betti functions from a suite of N-body simulations and use these to train a Gaussian process emulator that provides rapid model predictions; we next run a Markov chain Monte Carlo analysis on independent mock data to infer the cosmological parameters and their uncertainties. When comparing our results, we recover the input cosmology and achieve a constraining power on S8 ≡ σ8Ωm/0.3 that is 3% tighter than that on peak count statistics. Performing the same analysis on 100 deg2 of Euclid-like simulations, we are able to improve the constraints on S8 and Ωm by 19% and 12%, respectively, while breaking some of the degeneracy between S8 and the dark energy equation of state. To our knowledge, the methods presented here are the most powerful topological tools for constraining cosmological parameters with lensing data.

We show that the complex of free factors of a free group of rank n 2 is homotopy equivalent to a wedge of spheres of dimension n − 2. We also prove that for n 2, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent to the complex of free factor systems and moreover is (n − 2)-connected. In addition, we show that for every non-trivial free factor system of a free group, the corresponding relative free splitting complex is contractible.

We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen-Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi subgroups and describe higher generating families of parabolic subgroups in Aut(Fn).

We demonstrate how to use persistent homology for cosmological parameter inference in a tomographic cosmic shear survey. We obtain the first cosmological parameter constraints from persistent homology by applying our method to the first-year data of the Dark Energy Survey. To obtain these constraints, we analyse the topological structure of the matter distribution by extracting persistence diagrams from signal-to-noise maps of aperture masses. This presents a natural extension to the widely used peak count statistics. Extracting the persistence diagrams from the cosmo-SLICS, a suite of N -body simulations with variable cosmological parameters, we interpolate the signal using Gaussian processes and marginalise over the most relevant systematic effects, including intrinsic alignments and baryonic effects.For the structure growth parameter, we find S8 = 0.747 +0.025 −0.031 , which is in full agreement with other late-time probes. We also constrain the intrinsic alignment parameter to A = 1.54 ± 0.52, which constitutes a detection of the intrinsic alignment effect at almost 3σ.