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I'm reviewing Algebraic Topology and this is an old homework

"Viewing the torus T as usual as the square $[-1,1]^2$ with opposite sides identified, let $X$ be obtained from T by removing the open disk centered at the origin and with radius $1/2$. Find an explicit cell structure on $X$."

My solution: Let $a,b$ denote the sides of the squares. All the vertices of the square is a $0$-cell, say $x_0$. Let $x_1$ be a point on the circle whose interior is removed. Let $c$ denote a $1$-cell that connects from $x_0$ to $x_1$. Let $d$ denote a loop at $x_1$ (which represents the circle whose interior was removed). Then this is the cell structure: two $0$-cells $x_0,x_1$, four $1$-cells $a,b,c,d$ and one $2$-cell that attaches to $aba^{-1}b^{-1}cdc^{-1}$.

My questions:

  1. Is this a correct cell structure? My problem is that I think it is theoretically, but it's hard for me to imagine.

  2. Seifert - van Kampen theorem let us compute the fundamental group of a cell structure with one $0$-cell. So I cannot use it to compute this fundamental group. I know that geometrically speaking, $X$ can be deformation retracted to its boundary, which is a wedge of 2 circles and hence its fundamental group is $F_2$. I just wonder if we can find a cell structure that we can apply S-vK theorem to?

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    $\begingroup$ You can find a cell structure with one $0$ cell for any connected CW complex; it's just a matter of writing it down. $\endgroup$
    – pancini
    Aug 11, 2020 at 0:07
  • $\begingroup$ @ElliotG As far as i know, we can in general only find a homotopy equivalent CW-complex with the cell structure with a single $0$-cell using the construction of a CW-approximation as in Hatcher. Furthermore, i think that $I$ is an example of something that cannot be written with the cell structure with a single $0$-cell. We need at least two $0$-cells, because $I$ is not a wedge of spheres. $\endgroup$
    – Frederik
    Aug 11, 2020 at 6:50

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It is the correct cell structure. I have drawn a picture that might help to visualize it: enter image description here

2. We cannot in general give a cell structure for a connected CW-complex with a single $0$-cell. This is an example of this. To see this, note that the $1$-skeleton of our space is not a wedge of circles nor a single point, and the $1$-skeleton of a CW-complex with a single $0$-cell will either be a single point or a wedge of circles.

As you pointed out the CW-complex is homotopy equivalent to $S^1\vee S^1$ which can be given the structure with one $0$-cell. In fact, for any connected CW-complex $X$ we can find a CW-complex homotopy equivalent to it with the cell structure with a single $0$-cell. This can be read in Hacther's book in the section on CW-approximations.

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