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I would like to understand how to prove that the connected sum $\mathbb{R}P^2 \# T^2$ of the projective plane with a torus is homeomoprhic to $\mathbb{R}P^2 \# \mathbb{R}P^2 \# \mathbb{R}P^2$.

I got as far as showing that it must be equivalent to a connected sum of projective planes, how can I argue though that I need precisely three projective planes ?

Thanks for your help!

(P.S. not a homework exercise, this is for me to understand the classification of surfaces).

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2 Answers 2

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The following picture will help you to understand this problem intuitively. (If you can understand why $\mathbb{R}\mathrm{P}^2 \# \mathbb{R}\mathrm{P}^2 = K$, where $K$ stands for the Klein bottle.)

The figure in the upper right corner is $K \setminus \text{disk} $. It may take some time to think why it looks like this.

T^2 # \mathbb{R}\mathrm{P}^2 = \mathbb{R}\mathrm{P}^2 # \mathbb{R}\mathrm{P}^2 # \mathbb{R}\mathrm{P}^2

Source of the picture: link

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    $\begingroup$ Nice. Where do these pictures come from? $\endgroup$
    – PseudoNeo
    Commented Nov 15, 2013 at 16:45
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    $\begingroup$ @PseudoNeo, I'm sorry for not citing the source of the picture. I have added the link to my answer. $\endgroup$ Commented Nov 15, 2013 at 16:51
  • $\begingroup$ No worries. I was genuinely curious. $\endgroup$
    – PseudoNeo
    Commented Nov 15, 2013 at 16:54
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    $\begingroup$ These pictures originally come from the book "The shape of space" by Jeffrey Weeks. $\endgroup$
    – Nick Alger
    Commented May 16, 2017 at 1:54
  • $\begingroup$ Is the statement that a handle is equivalent to two cross-caps true in the case of surfaces with boundary? $\endgroup$
    – QGravity
    Commented Jun 3, 2019 at 2:56
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Calculate the Euler characteristic. This, combined with the fact that the resulting surface is non-orientable gives you the complete set of invariants, enough to single out the $\mathbb{R}P^2 \# \mathbb{R}P^2 \# \mathbb{R}P^2$.

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    $\begingroup$ I believe that you're "putting the cart before the horse" here. The OP's intent is to understand the classification of surfaces, one important step of which is to establish the stated homeomorphism. $\endgroup$ Commented Apr 11, 2013 at 20:19
  • $\begingroup$ Is the statement that a handle is equivalent to two cross-caps true in the case of surfaces with boundary? $\endgroup$
    – QGravity
    Commented Jun 3, 2019 at 2:56

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