Gluing semi-edges of a square starting from corners

Half side lengths of a unit square are glued together so that, just like corners, side centers are also becoming a point. Is it now homeomorphic to a sphere with a hole or to a no-holed un-punctured sphere?

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1 Answer
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This is homeomorphic to a sphere no holes assuming the gluing directions are “aa−1aa^{-1}” for each of the edges. You can take a piece of paper and identify the edges in the given manner. You’ll see that you haven’t introduced an extra hole.

Otherwise, you can compute the Euler characteristic. There are 5 vertices, 4 edges, and 1 face so χ=5−4+1=2\chi = 5 – 4 + 1 = 2, and classification of surfaces tells you this is a sphere.

If you glue according to “aa””aa”, then there is only 1 vertex, and the edges and faces remain the same so χ=1−4+1=−2\chi = 1 – 4 + 1 = -2 so you need to check whether or not it is orientable and classification of surfaces will give you your result.