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There are 2 projective planes. Each has 2 punctures. If the PPs are joined along the 2 punctures, what will be the Euler Characteristics of the new body?

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Each PP has two punctures.

PP1 has 2 punctures and PP2 also has 2. We now connect 1 puncture of PP1 to one of PP2 and the other puncture of PP1 to other of PP2

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  • $\begingroup$ You are using the terminology "puncture" rather loosely, leading to some imprecision in your question. I presume that a "puncture" is a point in the closed surface that is not in the punctured surface, which means that PP1 is homeomorphic to the projective plane minus two points, and that PP2 is also homeomorphic to the projective plane minus two points. If that presumption is correct, what does it mean to "connect 1 puncture of PP1 to one of PP2 and the other puncture of PP1 to other of PP2"? What kind of connection are you using? $\endgroup$
    – Lee Mosher
    Feb 17, 2017 at 14:40
  • $\begingroup$ I interpret puncture as a open 2-disk removed $\endgroup$
    – janmarqz
    Feb 17, 2017 at 23:18

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A projective plane with a puncture is a Möbius band, and two of them glued by their boundaries will be a Klein bottle, but the extra punctures glued apart will give you an extra handle in the Klein bottle, so your surface is $T\# K$, the genus four non orientable surface. From here you could say its Euler Characteristic.

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  • $\begingroup$ Each PP has two punctures. PP1 has 2 punctures and PP2 also has 2. We now connect 1 puncture of PP1 to one of PP2 and the other puncture of PP1 to other of PP2 $\endgroup$
    – zhirzh
    Feb 17, 2017 at 4:10
  • $\begingroup$ @zhirzh: right, you have a Klein bottle with two punctures but these identified, so an extra handle arises $\endgroup$
    – janmarqz
    Feb 17, 2017 at 4:12

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