  The abundance of peaks in weak gravitational lensing maps is a potentially
powerful cosmological tool, complementary to measurements of the shear power
spectrum. We study peaks detected directly in shear maps, rather than
convergence maps, an approach that has the advantage of working directly with
the observable quantity, the galaxy ellipticity catalog. Using large numbers of
numerical simulations to accurately predict the abundance of peaks and their
covariance, we quantify the cosmological constraints attainable by a large-area
survey similar to that expected from the Euclid mission, focussing on the
density parameter, {\Omega}m, and on the power spectrum normalization,
{\sigma}8, for illustration. We present a tomographic peak counting method that
improves the conditional (marginal) constraints by a factor 1.2 (2) over those
from a two-dimensional (i.e., non-tomographic) peak-count analysis. We find
that peak statistics provide constraints an order of magnitude less accurate
than those from the cluster sample in the ideal situation of a perfectly known
observable-mass relation; however, when the scaling relation is not known a
priori, the shear-peak constraints are twice as strong and orthogonal to the
cluster constraints, highlighting the value of using both clusters and
shear-peak statistics.
