  We present measurements of the two-dimensional genus of the SDSS-III BOSS
catalogs to constrain cosmological parameters governing the shape of the matter
power spectrum. The BOSS data are divided into twelve concentric shells over
the redshift range $0.2 < z < 0.6$, and we extract the genus from the projected
two-dimensional galaxy density fields. We compare the genus amplitudes to their
Gaussian expectation values, exploiting the fact that this quantity is
relatively insensitive to non-linear gravitational collapse. The genus
amplitude provides a measure of the shape of the linear matter power spectrum,
and is principally sensitive to $\Omega_{\rm c}h^{2}$ and scalar spectral index
$n_{\rm s}$. A strong negative degeneracy between $\Omega_{\rm c}h^{2}$ and
$n_{\rm s}$ is observed, as both can increase small scale power by shifting the
peak and tilting the power spectrum respectively. We place a constraint on the
particular combination $n_{\rm s}^{3/2} \Omega_{\rm c}h^{2}$ -- we find $n_{\rm
s}^{3/2} \Omega_{\rm c}h^{2} = 0.1121 \pm 0.0043$ after combining the LOWZ and
CMASS data sets, assuming a flat $\Lambda$CDM cosmology. This result is
practically insensitive to reasonable variations of the power spectrum
amplitude and linear galaxy bias. Our results are consistent with the Planck
best fit $n_{\rm s}^{3/2}\Omega_{\rm c}h^{2} = 0.1139 \pm 0.0009$.
