Let \(U=\{(x,y,0)\in\RR^3:x^2+y^2=1\}\), and let \(W=\RR^3-U\). Let \(V=\{(x,y,z)\in\RR^3:x^2+y^2+z^2=4\}\cup\{(0,0,z)\in\RR^3:-2\le z\le2\}\).
Describe a deformation retraction of \(W\) onto \(V\).
Use (a) and the Seifert–van Kampen Theorem to compute \(\pi_1(W)\).
Solution
Write points of \(\RR^3\) in cylindrical coordinates \((r,\theta,z)\) with \(r\ge0\). Rotation about the \(z\)-axis preserves \(U\) and \(V\), and a rotation orbit is determined by its image \((r,z)\) in the half-plane \(H=\{(r,z):r\ge0\}\). \(U\) is the orbit of \(p=(1,0)\), and \(V\) is the union of the orbits of \(\partial_HD=\{r^2+z^2=4\}\cup\{r=0,\ \abs{z}\le2\}\), the boundary of the closed half-disk \(D=\{(r,z)\in H:r^2+z^2\le4\}\) relative to \(\RR^2\); \(p\) lies in the interior of \(D\).
1. \(H\setminus\{p\}\) strongly deformation retracts onto \(D\setminus\{p\}\) by a homotopy preserving each ray from the origin.
For \(u\in H\) let \(q(u)=u\) if \(\norm{u}\le2\) and \(q(u)=2u/\norm{u}\) if \(\norm{u}>2\), and put \(Q_s(u)=(1-s)u+sq(u)\). \(Q_s\) fixes \(D\) and moves a point outside \(D\) along its ray while keeping its norm at least \(2\), so it never meets \(p\), whose norm is \(1\).
2. \(D\setminus\{p\}\) strongly deformation retracts onto \(\partial_HD\) by a homotopy fixing \(\partial_HD\) pointwise.
\(D\) is compact and convex with \(p\) in its interior, so each ray from \(p\) meets \(\partial_HD\) in exactly one point \(\rho(u)\), and \(\rho\colon D\setminus\{p\}\to\partial_HD\) is continuous. \(R_s(u)=(1-s)u+s\rho(u)\) stays on the ray from \(p\) through \(u\), so it never meets \(p\), and \(\rho(u)=u\) for \(u\in\partial_HD\).
3. \(W\) strongly deformation retracts onto \(V\).
Apply \(Q_s\) and then \(R_s\) to \((r,z)\) while keeping \(\theta\) fixed. \(Q_s\) preserves rays from the origin, so points on the \(z\)-axis stay on it; \(R_s\) fixes the diameter \(\{r=0,\ \abs{z}\le2\}\) pointwise, and a point with \(r>0\) reaches \(r=0\) only at \(s=1\). So the lifted homotopies are well defined and continuous on \(W\), fix \(V\) pointwise, and end with image the rotation of \(\partial_HD\), which is \(V\).
4. \(\pi_1(V)\cong\ZZ\).
Let \(S\) be the sphere of radius \(2\) and \(I\) the polar segment. A meridian arc \(\mu\subset S\) joining the poles is contractible, and \(V/\mu\cong S^2\vee S^1\): the sphere becomes a sphere, and \(I\), whose endpoints are identified, becomes a circle attached at one point. Collapsing the contractible subcomplex \(\mu\) is a homotopy equivalence, so \(V\simeq S^2\vee S^1\). By the Seifert–van Kampen theorem for wedge sums, \(\pi_1(S^2\vee S^1)\cong\pi_1(S^2)*\pi_1(S^1)\cong\ZZ\).
5. Q.E.D.
Step 3 answers (a). For (b), the deformation retraction of step 3 induces an isomorphism \(\pi_1(W)\cong\pi_1(V)\), so by step 4, \(\pi_1(W)\cong\boxed{\ZZ}\).