  We link the structure of nuclei around $^{100}$Sn, the heaviest doubly magic
nucleus with equal neutron and proton numbers ($N=Z=50$), to nucleon-nucleon
($NN$) and three-nucleon ($NNN$) forces constrained by data of few-nucleon
systems. Our results indicate that $^{100}$Sn is doubly magic, and we predict
its quadrupole collectivity. We present precise computations of $^{101}$Sn
based on three-particle--two-hole excitations of $^{100}$Sn, and reproduce the
small splitting between the lowest $J^\pi=7/2^+$ and $5/2^+$ states. Our
results are consistent with the sparse available data.
