I have a question regarding fundamental groups.
If I take a sphere and union a line between it's poles, is that the same space as the sphere with those poles identified? I am trying to find the fundamental group of the former, and know how to find the fundamental group of the latter, and I also know they have the same fundamental group. Is this the reason why?
I have yet to see a concise solution for the first one on the internet, so I am wondering if this more general statement about lines vs identifications of points is true.
I suspect you are asking about the following picture:
which is (a coloured version of) Figure 7.9 in my book Topology and Groupoids where the relation with a result on mapping cylinders is explained. That result is that if $i: A \to X$ is a closed cofibration, and $f: A \to B$ is a map, then the mapping cylinder $M(f)\cup X$ is homotopy equivalent to the adjunction space $B \cup_f X$. For your example, $X$ is the $2$-sphere, $S^2$, $A$ consists of the North and South poles, $B$ is a single point. The left hand figure is the adjunction space, and the right hand figure is mapping cylinder.
I also think I have given this picture elsewhere on this site or mathoverflow!