Connected Sum of Klein Bottle and Projective Plane

Question

I am asked to derive polygon and gluing scheme for connected sum of Klein Bottle, $K^2$ and Projective Plane,$P^2$, $K^2 \# P^2$ -not caring orientation issues-. Is it possible for me to cut anywhere of each polygons? Below you can find my attempt. Is it correct?

Answer 1

No, you are not allowed to cut anywhere you want, and in particular you are not allowed to cut along the red arcs in your two diagrams.

Let's review the definition of connected sum of two surfaces (not caring about orientation issues):

1. Cut each of the two surfaces along an embedded circle that is the boundary of an embedded disc, and throw away the interiors of the two discs. In what remains of each surface there is now a boundary circle.
2. Glue those two boundary circles, using any chosen boundary homeomorphism between those two circles.

What you have violated is that you may only cut along an embedded circle (and only one that is the boundary of an embedded disc). In each diagram you have cut along an arc, and that would be okay IF that arc became an embedded circle after gluing (and if that circle bounded an embedded disc). However, you should be able to see that the endpoints of your red arc are not identified to the same point after gluing, and so that red arc does not become a circle after gluing, it stays as an arc.

What I suggest you do instead is to draw a red circle in the middle of each diagram, cut along that circle, and then throw away the inner disc that circle bounds. Your two diagrams will now have holes in the middle, and so your two squares will have become "square annuli". To proceed furthermore, you may wish to do more cut operations before you start re-pasting, i.e. make a cut from the inside to the outside of each square annulus to make two new polygons.

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By the way, your second picture is not a Klein bottle: it's another projective plane.