[Math] how should I show that it is wedge of infinite circles

Question

I know that the shape that we see it below is homotopy equivalent of wedge of infinite circles,so the fundamental group of it is $\prod _{1}(\vee _{\alpha \in A}S^{1})=\ast _{\alpha \in A }\mathbb{Z}$,but I don't know how should I show that it is wedge of infinite circles,also I can't imagine what is happening for this shape.please help me with your knowledge,thanks.

Answer 1

This is a difficult homotopy equivalence to visualize, but I've attempted to draw a picture of what's going on. First you poke a hole in your surface starting at $+\infty$ and pushing in from the right. Similarly poke a hole from the left. You can see this is a homotopy equivalence by analogy with an infinite cylinder, which can be visualized as having two dotted boundary components at $\pm\infty$. This transforms your surface into an infinite strip with a band connecting top and bottom and infinitely many tubes. Now look at the righ-hand side of my picture which shows a square with a tube in the middle. Once you draw in the indicated $1$-cells, the complement is a disk, which you can use to push the top boundary onto the remaining $1$-cells as indicated in the lower right. Do this for infinitely many squares in a row on your strip all at the same time, to get the bottom left picture. From here, it is easy to see this is a wedge of infinitely many circles. Note that you get one "extra" circle, besides the obvious infinitely many pairs.