We report on the results from H{\alpha} imaging observations of the eastern limb of Tycho's supernova remnant (SN1572) using the Wide Field Planetary Camera 2 on the Hubble Space Telescope. We resolve the detailed structure of the fast, collisionless shock wave into a delicate structure of nearly edge-on filaments. We find a gradual increase of H{\alpha} intensity just ahead of the shock front, which we interpret as emission from the thin (~1") shock precursor. We find that a significant amount of the H{\alpha} emission comes from the precursor and that this could affect the amount of temperature equilibration derived from the observed flux ratio of the broad and narrow H{\alpha} components. The observed H{\alpha} emission profiles are fit using simple precursor models, and we discuss the relevant parameters. We suggest that the precursor is likely due to cosmic rays and discuss the efficiency of cosmic-ray acceleration at this position.