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Let $G=\mathbb{Z}\oplus \mathbb{Z}$ with generators $a,b \in G$. We can define cell complexes $[a,b]$ and $[a|b]$ as I did in the images below. For example, $[a,b]$ is simply the 2-simplex with edges labelled by $[a]$, $[b]$ and $[ab]=[ba]$ and vertices labelled by $[\;]$.

However, in the end I would like to identify cells with the same label/color (respecting the orientation!). A reversed orientation is indicated by a minus sign. For $[a,b]$, $[a|b]$ and $[a,a]$ you can see the result on the right hand side.

Note that $[a|b]$ is the solid torus, for simplicity I drew just its boundary.

Now, I'm not very good in topology, so I have two questions:

  1. What kind of 3-fold do we get after identifying cells of the cell complex $[a|a]$ (see picture below)?
  2. What kind of 3-fold do we get after identifying cells of the cell complex $[a|b]+[b|a]$ (see picture below)?

EDIT: I believe that $[a|a]$ and $[a|b]+[b|a]$ are lens spaces since we glue solid tori together, but I'm not sure which ones.

Please feel free to ask, if some part of my question is too vague. I hope that the pictures are self-explanatory, but maybe I'm wrong.

enter image description here enter image description here

EDIT: I added a picture of $[a,a]$ above. Here's another one:

enter image description here

It's quite easy to see how to turn it over, but then I don't understand how to glue both copies together.

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    $\begingroup$ The pictures look nice but I still cannot understand what you mean. For example do you glue $[a, b]$ with $-[a, b]$ after flipped? If so, then it seems that there is no difference between $[a|b] + [b|a]$ and $[a|b]$. Also I suppose $3$-fold usually means something of dimension $3$, but here you seem to have only two dimensional objects. Perhaps you could write down a more rigorous construction to clarify some points. $\endgroup$
    – WhatsUp
    Commented Dec 7, 2021 at 17:57
  • $\begingroup$ @WhatsUp Thx for your comment! What I drew is indeed a bit missunderstandable: $[a|b]$ is the $\textbf{solid}$ torus with boundaries $[a,b]$ and $-[b,a]$, so its a 3-fold. And you're right, we glue $[a,b]$ with $-[a,b]$ after flipping, but $[a|b]+[b|a]$ can't be the same as $[a|b]$, since it must contain two solid tori and not one. $\endgroup$
    – Bipolar Minds
    Commented Dec 7, 2021 at 18:21
  • $\begingroup$ I added a picture of $[a,a]$. It's quite easy to see how to turn it over, but then I don't understand how to glue both copies together. $\endgroup$
    – Bipolar Minds
    Commented Dec 8, 2021 at 10:58
  • $\begingroup$ I'm also a bit confused by the definition of $[a\mid b]$. The square that you have before identifying things, being a square, is 2-dimensional. How do 3-cells arise in this picture? Do you by any chance have an explicit description of the cells involved in its construction? $\endgroup$
    – Jeroen van der Meer
    Commented Dec 9, 2021 at 18:51
  • $\begingroup$ @JeroenvanderMeer That's a really good question. To be honest, I don't know how to describe $[a|b]$ as a cell complex before the identification. You're right that as it stands, it only makes sense as a 3-fold after identification. One idea was to introduce an additional 2-cell $[a \wr b]$ being just a non-tiled square face with outer edges being the same as those of $[a,b]-[b,a]$. Then we could glue these outer edges of $[a \wr b]$ and $[a,b]-[b,a]$ together and fill this object with a 3-cell. $\endgroup$
    – Bipolar Minds
    Commented Dec 9, 2021 at 19:48

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