(i) Define manifold and manifold with boundary for subsets $X \subset \mathbb{R}^{N}$.

(ii) Let $X$ and $Y$ be manifolds and $f: X \rightarrow Y$ a smooth map. Define what it means for $y \in Y$ to be a regular value of $f$.

(iii) Let $n \geqslant 0$ and let $\mathbb{S}^{n}$ denote the set $\left\{\left(x^{1}, \ldots, x^{n+1}\right) \in \mathbb{R}^{n+1}: \sum_{i=1}^{n+1}\left(x^{i}\right)^{2}=1\right\}$. Let $B^{n+1}$ denote the set $\left\{\left(x^{1}, \ldots, x^{n+1}\right) \in \mathbb{R}^{n+1}: \sum_{i=1}^{n+1}\left(x^{i}\right)^{2} \leqslant 1\right\}$. Show that $\mathbb{S}^{n}$ is an $n$-dimensional manifold and $B^{n+1}$ is an $(n+1)$-dimensional manifold with boundary, with $\partial B^{n+1}=\mathbb{S}^{n}$.

[You may use standard theorems involving regular values of smooth functions provided that you state them clearly.]

(iv) For $n \geqslant 0$, consider the map $h: \mathbb{S}^{n} \rightarrow \mathbb{S}^{n}$ taking $\mathbf{x}$ to $-\mathbf{x}$. Show that $h$ is smooth. Now let $f$ be a smooth map $f: \mathbb{S}^{n} \rightarrow \mathbb{S}^{n}$ such that $f \circ h=f$. Show that $f$ is not smoothly homotopic to the identity map.