We investigate the volume transition of a thermoresponsive polymer, poly(N-isopropylacrylamide), PNIPAM, in the presence of an aqueous solution of nonionic surfactant, C(12)E(9). 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 C(12)E(9) 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 C(12)E(9), even for relatively high concentrations of C(12)E(9). 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 C(12)E(9) 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.