Abstract　 　　
　Stars are assembled in molecular clouds where over density regions collapse as gravity overcomes the internal pressure. Magnetic fields and turbulence are the two possible forces that counteract gravity and hinder star formation. While the role of turbulence in this process is well recognized, how magnetic fields may influence the dynamics has been the subject of vigurous debate for decades. We investigate the role of magnetic fields in massive star and cluster formation through images of dust polarization at 870 micron obtained with the Submillimeter Array (SMA). The sample of 14 star forming clumps contains $10² - 10³$ \msun\ of molecular gas within a size of <~1pc at average densities > 10⁴$ cm$^{-3}$. The SMA observations reveal a polarization pattern in these objects ranging from ordered hour-glass configurations to more chaotic distributions. By comparing polarizations in the $10⁴$ cm$^{-3}$ dense gas with the parsec-scale polarization obtained with single dish telescopes, we found that magnetic fields in the majority of the dense gas are aligned within $40^\circ$ of the parsec-scale magnetic fields. With this unprecedentedly large sample, we argue on a statistical basis that magnetic fields play an important role during the collapse and fragmentation of massive molecular clumps and the formation of dense cores.