We investigate the volume transition of a thermoresponsive polymer, poly(N-isopropylacrylamide), PNIPAM, in the presence of an aqueous solution of nonionic surfactant, C12E9. We combine turbidimetry with optical microscopy, NMR, and SAXS to follow the volume transition of the PNIPAM and the H-1-isotropic transition of the surfactant/water system. Nonionic surfactants such as C12E9 are known to interact weakly with PNIPAM. Accordingly, we show that there is only a small change in the volume transition temperature for the PNIPAM in isotropic micellar solutions of C12E9, even for relatively high concentrations of C12E9. Interestingly, once the surfactant forms an H-1 phase, there is a dramatic decrease in the coil globule transition onset temperature. We believe that this behavior results from a competition between C12E9 in the H-1 phase, and PNIPAM to associate with water. When PNIPAM in the H-1 phase is cooled to low enough temperatures so as to be in the coil state, it locally disturbs the hexagonal phase ordering. Thus, we show that for PNIPAM in a weakly interacting surfactant matrix, it is the phase behavior of the matrix rather than the matrix chemistry that governs the coil globule transition. Finally, we show that in a PNIPAM copolymer with a higher LCST we observe an interesting sequence of transitions in the surfactant phase: on cooling from a high temperature free-blowing turbid globular state (similar to 75 degrees C), we enter a free-flowing translucent coil phase (similar to 47 degrees C), then a turbid gel (similar to 25 degrees C) where the copolymer is collapsed in the HI phase, and finally a low-temperature clear gel (similar to 5 degrees C) where the copolymer is in the expanded coil state.