CW complex structure on standard sphere identifying the south pole and north pole I need to find CW complex structure in the sphere by identifying the north and south pole.
First of all I tried to visualize what it looks like and if I am not wrong gluing the poles together will result in something looking sort of like a torus but without a hole. Now from a question that another user has posed namely(Cell Structure on Sphere with Two points identified) I saw that the CW complex structure is the following attach the boundary of the one cell to the zero cell and then attach the boundary of the two cell to the zero cell. To me this structure results to a wedge of a sphere and a circle because attaching the one cell to the zero cell gives a circle and then attaching the two cell to the zero cell will result to a sphere. As such this structure will result to the wedge of a sphere and a circle. Can you please explain to me what's wrong with my reasoning and how the structure above actually results to the sphere with the poles identified??
Thanks in advance
 A: Start with a 0-cell $x$. 
Attach an oriented 1-cell $e$ with both endpoints identified to $x$. 
Attach a 2-cell $\sigma$ with attaching map defined on its oriented boundary circle $\partial\sigma$ as follows:


*

*Subdivide $\partial\sigma$ as a concatenation of two arcs $\partial\sigma = \partial_1 \sigma * \partial_2 \sigma$, 

*Map $\partial_1 \sigma$ to $e$, 

*Map $\partial_2 \sigma$ to $\bar e$.


ADDED LATER: To see why this CW complex $X$ is homeomorphic to the sphere with north and south pole identified, this is the case where a picture is worth a large number of words, in order to convey the intuition. Yet it can be done with intuitive rigor also, with a few carefully chosen words, which I will attempt to supply.
First note that the characteristic map of the 2-cell $f : \sigma \to X$ is a surjective continuous map and is therefore a quotient map. So, $X$ can be reconstructed by starting with the 2-dimensional disc $\sigma$ and making the same identifications on $\sigma$ as are made by the map $f$. What are these identification? They are of two different types:


*

*Identify the two oriented arcs $\partial \sigma_1$, $\partial\bar\sigma_2$ to a single arc (which will be the 1-cell of $X$). 

*Identify the two points $\partial(\partial \sigma_1) = \partial(\partial\sigma_2)$ to a single point (which will be the $0$-cell of $X$).


So, how do we see that this is the 2-sphere with the north and south poles identified? First do just identification 1: the quotient is the 2-sphere, and the arcs $\partial\sigma_1$, $\partial\sigma_2$ each map to a longitude line connecting the north and south poles. Next do identification 2: identify the north and south poles.
