# Cell structure of a torus with an open disk removed

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?

• You can find a cell structure with one $0$ cell for any connected CW complex; it's just a matter of writing it down. Aug 11, 2020 at 0:07
• @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. Aug 11, 2020 at 6:50

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.