Recall that $\mathbb{R} P^{n}$ is real projective $n$-space, the quotient of $S^{n}$ obtained by identifying antipodal points. Consider the standard embedding of $S^{n}$ as the unit sphere in $\mathbb{R}^{n+1}$.

(a) For $n$ odd, show that there exists a continuous map $f: S^{n} \rightarrow S^{n}$ such that $f(x)$ is orthogonal to $x$, for all $x \in S^{n}$.

(b) Exhibit a triangulation of $\mathbb{R} P^{n}$.

(c) Describe the map $H_{n}\left(S^{n}\right) \rightarrow H_{n}\left(S^{n}\right)$ induced by the antipodal map, justifying your answer.

(d) Show that, for $n$ even, there is no continuous map $f: S^{n} \rightarrow S^{n}$ such that $f(x)$ is orthogonal to $x$ for all $x \in S^{n}$.