= Solution
The homotopy lemma says that homotopic smooth maps between compact connected manifolds have the same mod-two degree, computed as the parity of the inverse image of a regular value. The homogeneity lemma says that any two points of a connected smooth manifold are related by a diffeomorphism isotopic to the identity.
Given regular values $p,q$ of $f$, choose such a diffeomorphism $h$ with $h(p)=q$. Then $h\circ f$ is homotopic to $f$, while $(h\circ f)^{-1}(q)=f^{-1}(p)$. The homotopy lemma proves that the two inverse-image counts have equal parity.
For smooth Brouwer, suppose a smooth self-map of a ball had no fixed point. Following the ray from $f(x)$ through $x$ to the boundary constructs a smooth retraction of the ball onto its sphere. Its restriction to the sphere is the identity, but it is also null-homotopic through the ball. The identity has odd mod-two degree and a constant map has even degree at a different regular value, contradicting the homotopy lemma. Thus a fixed point exists.
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