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Taking a square or rectangle and enforcing periodic boundary conditions (i.e. identifying opposite edges) forms a torus, which is a genus 1 manifold. What sort of torus-like structure is formed if we use a different polygon as our starting point, such as a hexagon? The polygon certainly must be able to tile a plane for the construction to work, but doesn't need to be regular. As an extension it would be interesting to know what happens if we enforce anti-periodic boundary conditions instead. In the case of a rectangle, mixing periodic and anti-periodic boundaries gives the Klein bottle.

Edit: My primary interest was in the context of the First Brillouin Zone of a 2D lattice (the Wigner Seitz cell of the reciprocal lattice). In condensed matter this concept is ubiquitous and in general (in 2D) it will be some polygon with periodic boundary conditions. For a square or rectangular lattice, it is obvious that it must form a torus, but it seems less obvious to me that when it is, for example, a hexagon, the FBZ will still be a torus.

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    $\begingroup$ There is an extensive and detailed discussion of this in Massey's book on algebraic topology. It explains how to take the information about which edges of the polygon are matched up, use that to construct a certain graph and then turn the graph into a formal description of the fundamental group of the manifold. When the manifold is a torus, the fundamental group is $\Bbb Z^2$; when it's a sphere the group is trivial. $\endgroup$
    – MJD
    Nov 13, 2019 at 16:03
  • $\begingroup$ If you could say with more mathematical precision what you regard as a "torus-like structure" that would help, otherwise your answerers are reduced to guessing or interpreting what you might mean. $\endgroup$
    – Lee Mosher
    Nov 16, 2019 at 21:14
  • $\begingroup$ @LeeMosher thanks I added a clarification $\endgroup$
    – Kai
    Nov 17, 2019 at 1:47

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I'll interpret your question as referring to topological manifolds.

It is possible to construct any compact surface by gluing edges together. An efficient way of representing them is to start with a closed disc $D_2$ with a chosen orientation on its boundary, and partition the circular boundary into $2n$ segments. Going around the circle in order, we can label the segments with integers $a_1,a_2,\dots a_n\in\mathbb{Z}$ such that for each integer $m\in[1,n]$, either $m$ appears exatly twice and $-m$ not at all, or $m$ and $-m$ each appear exatly once. Then we can glue segment $m$ to segment $\pm m$, with the minus sign denoting a flipped orientation relative to the boundary. We can write any identification of a polygon this way, though the labeling is of course not unique, and some labelings may not result in well defined quotient spaces.

As an example, for 4 segments, we have 3 cases: The sphere $(1,-1,2,-2)$, the torus $(1,2,-1,-2)$, and the real projective plane $(1,2,1,2)$. For a hexagon, things look the same; at least one pair of edges will be redundant.

More generally, we can take connected sums of these simple surfaces by concatenating their sequences, shifting indices as needed. It turns out every compact surface can be written as such a connected sum. As an example, connecting two tori gives an oriented genus two surface $(1,2,-1,-2,3,4,-3,-4)$.

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