= Solution
A <galaxy cluster> deflects light from background galaxies according to its projected gravitational field. In a thin gravitational lens approximation,
$$
\boldsymbol\beta=\boldsymbol\theta-\boldsymbol\alpha(\boldsymbol\theta),\qquad
\kappa=\frac{\Sigma}{\Sigma_{\rm crit}},\qquad
\Sigma_{\rm crit}=\frac{c^2D_s}{4\pi GD_lD_{ls}}.
$$
The <distances> are angular-diameter <distances>; only <mass> in the appropriate foreground/background geometry lenses the chosen source this way. Near critical curves, <strong gravitational lensing> produces highly magnified arcs and multiple images of the same source. Farther out, <weak gravitational lensing> produces small coherent shear distortions, inferred statistically because individual galaxies have unknown intrinsic ellipticities. The measured <reduced shear> is $g=\gamma/(1-\kappa)$. Transparent lensing obeys <surface-brightness conservation in gravitational lensing> at fixed source redshift while changing solid angle and integrated flux.
Strong-lensing image positions and source redshifts constrain the central projected <mass>; weak shear extends the map to larger radii. Multiple source planes help constrain geometry and degeneracies, but the <mass-sheet degeneracy> illustrates why image shapes alone do not uniquely fix absolute <mass>. Point-spread-function anisotropy, selection, shape calibration, cluster-member contamination, source redshift errors and unrelated line-of-sight structure require controls. Lensing does not require hydrostatic equilibrium, making it valuable for disturbed clusters, although interpretation still depends on the gravitational model.
An instructive observation is the merging <Bullet Cluster>: weak-lensing <mass> concentrations approximately follow the collisionless galaxies while the dominant visible baryonic <plasma>, traced by <X-rays>, is displaced. Clowe and collaborators combined ground-based and Hubble images for lensing reconstruction and compared them with the <gas> distribution, providing evidence for <mass> beyond the observed <baryons>.<A direct empirical proof of the existence of dark matter by Douglas Clowe and others>[abstract and mass-map/gas comparison]{ref}
\b[Cluster lensing maps total projected gravitating <mass> and magnifies background sources; it does not by itself identify the microscopic dark-matter particle.]
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