According to the Mechaber, IIRC, only wool and linen require tsitsit on a Torah level, cotton midirabanan, leather nothing. According to the Rema, cotton is also medioryta.
I believe that the Mechaber is paskening like the Rambam. I beleive the following are proofs for wool and linen are the only Torah begadim. When the Torah speaks beged by tzaraat, it mentions wool and linen. Furhtermore, in Devarim, when the Torah mentions shaatnez (wool and linen) is next to where it talks about Tsitsit (called there gedillim).