A polymer having the following monomers in polymerized form (in weight %): A) 49-60% of a first monomer which is selected from the group consisting of methyl acrylate, methyl methacrylate, ethyl acrylate, ethyl methacrylate, C3-alkyl acrylate, C3-alkyl methacrylate, C4-alkyl acrylate, C4-alkyl methacrylate and mixtures thereof, B) 30-40% methacrylic acid, C) 4-15% acrylic acid, and D) 0.02-0.30% of a crosslinking agent having at least two allyl moieties, wherein the sum of the amounts of monomers A to D is 100%. A hair styling composition containing the polymer and use of the polymer or of the hair styling composition for styling hair also are disclosed.