3
$\begingroup$

It is well known that if you start with a two-dimensional square and could glue — in the most straightforward way — the top and bottom edges to each other, and likewise the left and right edges to each other, then the resulting surface is a torus.

Let the square be the points (x,y) of the plane with 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.

Suppose instead that we glue the point (x,0) to the point (x',1) where x' = x + 1/2 (modulo 1), for all x with 0 ≤ x ≤ 1. And likewise, glue the point (0,y) to the point (1,y'), where y' = y + 1/2 (modulo 1), for all y with 0 ≤ y ≤ 1.

What surface is the result, topologically?

(Note that for any number t with 0 ≤ t ≤ 1, the expression

t + 1/2 (modulo 1)

means t + 1/2 if t < 1/2, and it means t - 1/2 if t ≥ 1/2.)

$\endgroup$
4
$\begingroup$

This is equivalent to gluing opposite sides of an octagon in an orientable manner. The result is a double torus (genus 2), as shown in the picture below taken from this MSE question: All segments meet at one point on the central handle

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.