# Proof of Brouwer fixed-point theorem using Brouwer-Poincaré theorem

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!

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.