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As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy theory or if it has at least some applications in algebraic topology! you can refer me to some refernces.. thanks!

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See: mathoverflow.net/questions/22294/… . –  Justin Young May 24 '13 at 13:06
    
@Rahman Have you get the version of Kakutani fixed point theorem in algebraic topology? –  Belle-tiantian Apr 15 at 3:52

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