Suppose there exists two copies of $S^1$, and 2 points on each $S^1$, say $a_1$ and $b_1$ on one copy and $a_2$ and $b_2$ on the other copy. If the points $a_i's$ are identified and $b_i's$ are identified with each other, then what will be the fundamental group of the resulting space.
I tried attaching the points $a_i$ with a line say A, and $b_i$ with a line say B. Then since this is a contractible $X\simeq X/A$. This gives $S^1 \vee S^1$ with two points from each circle identified. I would appreciate any help on how to proceed from here