The first systematic observations of sunspot umbrae using the lines of the Ti Ii multiplet at 2.2 mu m are presented. Their diagnostic capabilities are investigated, developed and used to investigate the magnetic and velocity structure of a sunspot. These lines are most sensitive to cool plasma. In addition, they are extremely Zeeman sensitive. We find that a sunspot is composed of two distinct cool magnetic components. One of them is fairly vertical, has a large magnetic field strength and is associated with the central (umbral) part of the sunspot. The other component is strongest near the outer boundary of the spot (penumbra), is much more inclined, has a very low magnetic field strength and shows the signature of the Evershed effect. In contrast to the smooth transition of field strength from the darkest part of the umbra to the outer penumbral boundary usually visible in observations carried out in other spectral lines, the Ti Ii lines exhibit a sharp transition between the two magnetic components.