”figure 8” space embedded in S2

Question

Let M3 be the 3-manifold defined as the quotient space of I × S2 by the identification {0} × {x} s {1} × {Tx}, where T : S2 → S2 is a reflection through a plane in R3. Find π1(M) and π2(M).

Answer 1

The universal cover is $\tilde M=\mathbb R\times S^2$: your manifold is $(\mathbb R\times S^2)/\mathbb Z$, with $n\in\mathbb Z$ acting by $n\cdot (t,x)=(t+n,T^nx)$. As the result, $\pi_1(M)=\mathbb Z$, $\pi_2(M)=\pi_2(\tilde M)=\mathbb Z$.