Real Projective Plane with One Handle:

The real projective plane with one handle is the unique non-orientable surface with Euler characteristic equal to -1.

Alternatively, it is the connected sum of a real projective plane with a torus. Less formally, it is a projective plane with two disks removed and a tube inserted between the two disks. This is demonstrated in the diagram below:

This is the triangulation of the projective plane with one handle used to build the tight polyhedral immersion presented here [More]. The two removed disks are shown in grey, and the tube is shown at the right. The dotted line is the double curve of the tight immersion.

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8/13/94 -- The Geometry Center