Proposition

Construct an explicit deformation retraction of $\mathbb{R}^n \setminus \{ 0 \}$ onto $S^{n - 1}$.

Solution

Let $F: \mathbb{R}^n \setminus \{ 0 \} \times I \rightarrow S^{n - 1}$ be defined such that

\[\begin{align*} F((x_1, \cdots, x_n), t) = \frac{(x_1, \cdots, x_n)}{(1 - t) + td} \end{align*}\]

where $d = \sqrt{x_1^2 + \cdots + x_n^2}$.

  • $F$ is continuous since $(1 - t) + td$ is continuous and a quotient of a continuous function by a continuous function is continuous.
  • $F(x, t) = x$ for any $x \in S^{n - 1}$ and $t \in I$.
  • $F(x, 1) \in S_1$ for any $x \in S^{n - 1}$.

Therefore, $F$ is a deformation retraction of $\mathbb{R}^n \setminus \{ 0 \}$ onto $S^{n - 1}$.