Let \(X\) be the quotient space of \(S^2\) obtained by identifying the north and south poles to a single point. Put a cell complex structure on \(X\) and use this to compute \(\pi_1(X)\).
Solution
Goal. Compute \(\pi_1(X)\) for \(X = S^2\) with north and south poles identified.
1. \(X\) is homeomorphic to \(S^1 \vee S^2\).
1.1. Identifying the two poles of \(S^2\) gives a space with a single “pinch” point.
the quotient collapses the two poles to one point.
1.2. This space is \(S^1 \vee S^2\) (a circle wedged with a sphere).
the arc from north to south pole becomes a circle (the two poles identified), and the rest of \(S^2\) becomes a sphere attached at the pinch point; the result is \(S^1 \vee S^2\).
2. Cell structure of \(S^1 \vee S^2\).
2.1. \(S^1\) has one \(0\)-cell and one \(1\)-cell; \(S^2\) has one \(0\)-cell and one \(2\)-cell.
standard cell structures.
2.2. \(S^1 \vee S^2\) has one \(0\)-cell (the wedge point), one \(1\)-cell, and one \(2\)-cell.
the two \(0\)-cells are identified at the wedge point.
3. Compute \(\pi_1(X)\).
3.1. \(\pi_1(S^1 \vee S^2) = \pi_1(S^1) * \pi_1(S^2) = \ZZ * 0 = \ZZ\).
van Kampen (or the fact that \(\pi_1\) of a wedge is the free product of the \(\pi_1\)’s), and \(\pi_1(S^2) = 0\).
3.2. Hence \(\pi_1(X) = \ZZ\).
1.2 and 3.1.
4. Q.E.D.
\(\pi_1(X) = \ZZ\).