Hatcher 0.2

Hatcher, Algebraic Topology, Chapter 0

2. Construct an explicit deformation retraction of \mathbb{R}^n\setminus\{0\} onto S^{n-1}.
Proof. This problem is essentially a problem from Calculus III. Note that for a vector \mathbf{x} in \mathbb{R}^n\setminus\{0\}, the normalized vector \mathbf{x}/\|\mathbf{x}\| lies on S^{n-1}. It suffices, then, to do the normalization process in a way that’s continuous for a time parameter t\in I, and one way to accomplish this is to define a family f_t:\mathbb{R}^n\setminus\{0\}\to S^{n-1} so that, for each \mathbf{x}=(x_1,x_2,\ldots,x_n) in the domain,

f_t(x_1,x_2,\ldots,x_n)=\left(\dfrac{x_1}{t\|\mathbf{x}\|+(1-t)},\cdots,\dfrac{x_n}{t\|\mathbf{x}\|+(1-t)}\right).

As noted in problem 1, the function f_t is continuous for each t\in I. Moreover, f_0(\mathbf{x})=\mathbf{x}, f_1(\mathbf{x})=\mathbf{x}/\|\mathbf{x}\|, and f_t(\mathbf{x})|_{S^{n-1}}=\mathbf{x} due to the fact that \|\mathbf{x}\|=1 for all \mathbf{x}\in S^{n-1}.    \,\,\square

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