I am trying to find a CW complex structure for $S^2$ with two poles identified. I would then use this to find its fundamental group, as in Hatcher 1.2.7. To solve this, I first tried to construct $S^1$ with two points identified. Here is my construction:
- Take two 0-cells, x and y
- Take three 1-cells, e.g. ]0, 1[, ]2, 3[ and ]4, 5[, and use attaching maps that identify 0 and y and 5 together, and 1 & x & 2 as well as 3 & x & 4 together. This gives the following diagram:[Diagram for $S^1$ with poles identified][1]
I think this gives what I want. I then attach two 2-cells to this circle to give the left and right hemispheres of a 2-sphere with 2 poles identified.
My questions:
- In all other questions on this topic, this is not the construction given. Is something wrong with my construction? Perhaps my attaching maps are not continuous, but I can't see why...
- Can I think of $S^1$ with two points identified as $S^1 \vee S^1$, i.e. a figure of eight? Intuitively, if I glue the two points together literally (thinking of the construction as embedded in $\mathbb{R}^2$) then I get a figure of eight. Are they homotopy-equivalent? [1]: https://i.stack.imgur.com/ocry3.jpg
