For each of these actions of \(\mathbb{Z}_2\) on \(S^n\), compute \(\pi_1(S^n/\mathbb{Z}_2)\)
\(S^1, z\mapsto -z\)
\(S^2, (x,y,z) \mapsto (-x,-y,z)\)
\(S^3, (z,w) \mapsto (-z, -w)\)
Solution
1. Case 1: \(S^1\) with \(z \mapsto -z\).
1.1. The action is free (no fixed points).
\(z = -z\) implies \(z = 0 \notin S^1\).
1.2. Hence \(S^1 \to S^1/\ZZ_2\) is a \(2\)-sheeted covering, and \(S^1/\ZZ_2 \cong S^1\).
the quotient of \(S^1\) by the antipodal map is again \(S^1\) (the map \(z \mapsto z^2\) identifies antipodal points).
1.3. Therefore \(\pi_1(S^1/\ZZ_2) = \pi_1(S^1) = \ZZ\).
2.
2. Case 2: \(S^2\) with \((x,y,z) \mapsto (-x,-y,z)\).
2.1. The fixed points are the two poles \((0,0,\pm 1)\).
\((-x,-y,z) = (x,y,z)\) iff \(x = y = 0\).
2.2. The quotient \(S^2/\ZZ_2\) is homeomorphic to \(S^2\).
Write a point of \(S^2\) as \((re^{i\theta},z)\in\CC\times\RR\) with \(r^2+z^2=1\). The action sends \((re^{i\theta},z)\) to \((re^{i(\theta+\pi)},z)\). Define \[ Q(re^{i\theta},z)=(re^{2i\theta},z). \] At the poles \(r=0\) this is independent of \(\theta\), and the formula is continuous there because the horizontal coordinate has norm \(r\). It is constant exactly on the two-point orbits of the rotation away from the poles and fixes each pole. Hence it factors through a continuous bijection \[ \overline Q:S^2/\ZZ_2\longrightarrow S^2. \] The domain is compact and the target Hausdorff, so \(\overline Q\) is a homeomorphism.
2.3. Therefore \(\pi_1(S^2/\ZZ_2) = \pi_1(S^2) = 0\).
2 and \(\pi_1(S^2) = 0\).
3. Case 3: \(S^3\) with \((z,w) \mapsto (-z,-w)\).
3.1. The action is free (no fixed points).
\((-z,-w) = (z,w)\) iff \(z = w = 0 \notin S^3\).
3.2. Hence \(S^3 \to S^3/\ZZ_2\) is a \(2\)-sheeted covering, and \(S^3/\ZZ_2 = \RP^3\).
the antipodal map on \(S^3\) has quotient the real projective space \(\RP^3\).
3.3. Therefore \(\pi_1(S^3/\ZZ_2) = \pi_1(\RP^3) = \ZZ/2\).
\(\pi_1(\RP^3) = \ZZ/2\) (its universal cover is \(S^3\) with deck group \(\ZZ/2\)).
4. Q.E.D.
\(\pi_1 = \ZZ\), \(0\), \(\ZZ/2\) respectively (<1>1, 2, <1>3).