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By double torus I mean the connected sum of two torus $\mathbb{T}$ denoted $\mathbb{T} \# \mathbb{T}$. By triple projective space I mean the triple connected sum of the projective plane $\mathbb{RP}_2$, denoted $\mathbb{RP}_2 \# \mathbb{RP}_2 \# \mathbb{RP}_2$. The group action that I use is given by the identity and the antipode homeomorphism is $A:\mathbb{T} \# \mathbb{T} \to \mathbb{T} \# \mathbb{T}$ given by $p \mapsto -p$.

I'm told to take as a fundamental region a disk in the "border" that connects both surfaces and study the action of the antipode on that disk. My intuition gives however only one $\mathbb{RP}_2$. What's going on?

An image of the alleged disk:

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Note that what I want to compute is $\mathbb{T} \# \mathbb{T}/ \{Id,A\}$.

Thoughts

Perhaps I could study first this antipode map for the simple torus $\mathbb{T}$ which gives the Klein bottle $\mathbb{RP}_2 \# \mathbb{RP}_2$. However, what result would allow me to generalize it for the double torus?

user1868607
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    one easy way to see is by computing Eular characteristic... – Anubhav Mukherjee Dec 21 '17 at 00:56
  • @AnubhavMukherjee Euler characteristic of the orbit space? how do you do that? – user1868607 Dec 21 '17 at 01:01
  • you can compute it as a connected sum – Anubhav Mukherjee Dec 21 '17 at 01:39
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    The notation $p\mapsto -p$ is meaningless unless you have in mind a specific embedding of your surface in a vector space. What you want to say is that you have a fixed-point free orientation-reversing involution of your surface. Then indeed the quotient is the connected sum of three projective planes. As Anubhav says, you can simply compute the Euler characteristic since for a degree $d$ cover $X\to Y$, $\chi(X)=d\chi(Y)$. – Moishe Kohan Dec 21 '17 at 02:37
  • https://math.stackexchange.com/questions/1227762/showing-the-sum-of-n-1-tori-is-a-double-cover-of-the-sum-of-n-copies-of-ma?rq=1 – Moishe Kohan Dec 21 '17 at 02:38

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