  The low-lying unbound level structure of the halo nucleus $^{19}\textrm{C}$
has been investigated using single-neutron knockout from $^{20}\textrm{C}$ on a
carbon target at 280 MeV/nucleon. The invariant mass spectrum, derived from the
momenta of the forward going beam velocity $^{18}\textrm{C}$ fragment and
neutrons, was found to be dominated by a very narrow near threshold
($E_\textrm{rel}$ = 0.036(1) MeV) peak. Two less strongly populated
resonance-like features were also observed at $E_\textrm{rel}$ = 0.84(4) and
2.31(3) MeV, both of which exhibit characteristics consistent with neutron
$p$-shell hole states. Comparisons of the energies, measured cross sections and
parallel momentum distributions to the results of shell-model and eikonal
reaction calculations lead to spin-parity assignments of $5/2^+_1$ and
$1/2^-_1$ for the levels at $E_x$ = 0.62(9) and 2.89(10) MeV with $S_n$ =
0.58(9) MeV. Spectroscopic factors were also deduced and found to be in
reasonable accord with shell-model calculations. The valence neutron
configuration of the $^{20}\textrm{C}$ ground state is thus seen to include, in
addition to the known $1s^2_{1/2}$ component, a significant $0d^2_{5/2}$
contribution. The level scheme of $^{19}\textrm{C}$, including significantly
the $1/2^-_1$ cross-shell state, is well accounted for by the YSOX shell-model
interaction developed from the monopole-based universal interaction.
