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We have just learned the proof of Brouwer fixed-point theorem(BFPT) using Lefschetz fixed-point theorem. And one of our homework is to show that Brouwer-Poincaré theorem implies BFPT. Brouwer-Poincaré theorem says that there is a nonvanishing continuous tangent vector field on $S^{n}$ if and only if n is odd. I understand fixed-point problem is in some sense equivalent to the existence of a null vector. But BFPT says nothing about the dimension n being even or odd. I can't see much connection between these two theorems. Could anyone help me? Thanks in advance!

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Hint: Any map $f:D^{2n-1} \to D^{2n-1}$ without a fixed point induces a map $g:D^{2n} \to D^{2n}$ given by $(\mathbf{x},y) \mapsto f(\mathbf{x},0)$ that has no fixed point.

If you want to take the idea further, the idea is to start with a fixed point free map $f$ and define a vector field $\mathbf{v}=\mathbf{x}-f(\mathbf{x})$. I think the details are a bit tricky.

Milnor takes this idea and proves the claim on the last pages of his paper here.

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