Is somebody able to prove that: If $f$ is a continuous mapping from $S^{2n-1}$ to $S^{2n-1}$ (the surface of a unit ball in $2n$ dimensional Euclidean space) and $f$ is not homotopic to the identity mapping, then $f$ has a fixed point.
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If $f$ has no fixed point then the Lefschetz number of $f$ is zero. But the Leftschetz number of $f$ is just $0 = Trf_0 - Trf_{2n-1} = 1 - Trf_{2n-1} = 1 - deg\,f$ which means that $deg\,f = 1$. But, for spheres, degree is a complete invariant so $f$ must be homotopic to the identity.
