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 $n_{rm s}^{3/2}Omega_{rm c}h^{2} = 0.1139 pm 0.0009$.
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