Fundamental Group of a sphere and a line joing the North and South poles

Question

Hi I am trying to compute the fundamental group of a sphere with a line through the centre joining the north and south poles. I initial thought is that it should be isomorphic to $\mathbb{Z}$ because if we pass through the line, we shouldn't be able to contract that loop. Am I correct, and if so, how would I go about formalising the argument?

Answer 1

This space is homotopy equivalent to $S^2$ with the north and south poles identified (collapse the line to a point.) In this case, it is homotopic to $S^1 \vee S^2$, which by Van Kampen's theorem has fundamental group $\mathbb Z$.

serr here for illustrative pictures! Or alternatively, chapter 0 of hatcher's online book also covers this example.