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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeJan 14th 2021

### As a homotopy type

As a homotopy type the torus is the product of two copies of the circle.

In homotopy type theory the torus can be formalized as the higher inductive type generated by a point base, two paths, $p$ and $q$, from base to itself, and an element $t$ of $p\cdot q = q \cdot p$. It has been formally shown (Sojakova15) that this type is equivalent to the product of two circles. For a treatment in cubical type theory, see (Licata-Brunierie).

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 4th 2022

added statement of the stable homotopy type of the 2-torus, and graphics illustrating this in line with the homotopy types of punctured tori (here)

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