bham10246
Jul14-07, 04:55 PM
Consider the following space X, consisting of two 2-spheres and two arcs glued together. Compute its fundamental group.
Since I can't draw a picture online, call the first sphere S_1 and call the second sphere S_2. Then one arc connects x_1 \in S_1 to x_2 \in S_2 and another arc connects y_1\in S_1 to y_2 \in S_2, where all the points are distinct.
I thought about this problem and contracted the arcs (so it looks like two 2-spheres identified in two points), and I want to use van Kampen. But I'm having a hard time figuring out two open sets A and B so that A \cup B = X and A\cap B is path connected.
Thank you!
Actually, can I contract one of the arcs so that the two 2-spheres touch at one point, then move the points x_1, x_2, y_1, y_2 to the point of intersection? Then I have S^2 \vee S^1\vee S^2. Then [itex]\Pi_1(X) = \mathbb{Z}[\itex]. Is this a correct analysis?
Since I can't draw a picture online, call the first sphere S_1 and call the second sphere S_2. Then one arc connects x_1 \in S_1 to x_2 \in S_2 and another arc connects y_1\in S_1 to y_2 \in S_2, where all the points are distinct.
I thought about this problem and contracted the arcs (so it looks like two 2-spheres identified in two points), and I want to use van Kampen. But I'm having a hard time figuring out two open sets A and B so that A \cup B = X and A\cap B is path connected.
Thank you!
Actually, can I contract one of the arcs so that the two 2-spheres touch at one point, then move the points x_1, x_2, y_1, y_2 to the point of intersection? Then I have S^2 \vee S^1\vee S^2. Then [itex]\Pi_1(X) = \mathbb{Z}[\itex]. Is this a correct analysis?