  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.
  We find for the structure growth parameter $S_8=0.747^{+0.025}_{-0.031}$,
which is in full agreement with other late-time probes. We also constrain the
intrinsic alignment parameter to $A=1.54\pm 0.52$, ruling out the case of no
intrinsic alignments at a $3\sigma$-level.
