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For some reason, I'm having trouble visualizing how to put a CW structure on a disk with 2 smaller disks removed. What I'd like to do is have three 0-cells, five 1-cells, and a single 2-cell. Three of the one cells would be glued to the three vertices to give three disjoint circles, and the other two 1-cells would be used to connect the three circles together. Then I can't see how to glue the 2-cell in, because when I visualize it, it always ends up covering one of the holes in the disk.

I realize my ability to visualize CW-complexes is not so good. For another example, it seems to me that if you take a wedge of two circles A and B, and glue a 2-cell on by $AB$, you should get a wedge of two disks, which would be contractible. But using cellular homology I see this has a nontrivial first homology group. I'd appreciate help sorting these two examples out to aid my intuition.

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  • $\begingroup$ A disk with 2 smaller disks removed has nontrivial first homology group $H_1(S)=\Bbb{Z}\oplus\Bbb{Z}$, being homotopy equivalent to the wedge of two circles, no? $\endgroup$
    – Managu
    Aug 3, 2012 at 1:25
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    $\begingroup$ Sure. The problem really is to put a CW structure on this, not to calculate anything. $\endgroup$
    – John
    Aug 3, 2012 at 1:27

2 Answers 2

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Please allow me to present my terrible GIMP sketch:

enter image description here

So we have points p1, p2, and p3, lines L1, L2, L3, L4, and L5, and I didn't label the obvious 2-cell. If I were to describe the attaching map of the 2-cell, I might start at p1, attach along L4, attach along L1, attach along -L4, attach along L5, attach along L2, attach along -L5, and attach along -L3 (ending at p1 again). I use +- to indicate direction, using the arrows on my rough sketch as the guide.

And as we can see - the fundamental group is very clearly nontrivial.

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Your thoughts on how to arrange the 0- and 1-cells are spot on. If you can draw the structure with just the 0- and 1-cells, you should see a good place to put the 2-cell. If you get stuck, I've drawn a picture:

https://i.stack.imgur.com/9zFDM.jpg

For the second example, recall that you're not actually filling in either of $A$ or $B$. Attaching a 2-cell via $AB$ means creating something like what you get when you pinch one of these:

https://i.stack.imgur.com/RYzAf.jpg

by holding a point on the top and bottom rings in your fist. Alternately, you can picture a piece of macaroni noodle with the smaller arc connecting the ends shrunk to a point. In fact, such a space is homotopy equivalent to a cylinder- simply unscrunch the join point of the circles into an interval. If you're curious about this sort of thing, some treatment is given in chapter 0 of Hatcher.

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  • $\begingroup$ Thank you, this was exactly what I had thought, but for some reason didn't think it worked. $\endgroup$
    – John
    Aug 3, 2012 at 1:46

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