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    Radial deformation retraction $\mathbb R^n\setminus\{0\}\to S^{n-1}$

    P-TOP-WORKSHOP-2020-WS2B-HW2

    Area
    Topology
    Topics
    Euclidean Spaces, Homotopy, Retracts
    Status
    draft

    Construct an explicit deformation retraction from \(\mathbb R^n\setminus\{0\}\) onto \(S^{n-1}\).

    Solution

    Define \[ H:(\mathbb R^n\setminus\{0\})\times[0,1]\to\mathbb R^n\setminus\{0\} \] by \[ H(x,t)=\left((1-t)+\frac{t}{\|x\|}\right)x. \] The scalar factor is positive for every \(x\ne0\) and \(0\le t\le1\), so \(H(x,t)\ne0\). Also \[ H(x,0)=x, \qquad H(x,1)=\frac{x}{\|x\|}\in S^{n-1}. \] If \(x\in S^{n-1}\), then \(\|x\|=1\), hence \[ H(x,t)=x \] for all \(t\). Thus \(H\) is a deformation retraction of \(\mathbb R^n\setminus\{0\}\) onto \(S^{n-1}\).

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