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Let $X$ be the quotient space obtained from an 8-sided polygonal region $P$ by pasting its edges together according to the labelling scheme $acadbcd^{-1}d$. a) Check that all vertices of $P$ are mapped to the same point of the quotient space $X$ by the pasting map. b) Calculate $H_{1}(X)$ c) Assuming $X$ is homeomorphic to one of the surfaces among $S^{2}, T_{1}, T_{2} \ldots, P_{1}, P_{2}, \ldots$, where $T_{n}$ and $P_{m}$ denote the $n$-fold connected sum of tori and the m-fold connected sum of projective planes, respectively, which surface is it?

I think I proved that $H_{1}(X)$ is $\mathbb{Z}_{2} * \mathbb{Z}^{3}$. However, I would like to have some feedback.

Thanks.

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    Could you expand a bit on what you have tried and what your thoughts are?2011-05-07
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    Usually in this and related settings star denotes free product of groups. But that isn't abelian, and homology is always abelian. Do you mean $\mathbb{Z}_2 \oplus \mathbb{Z}^3$?2011-05-08
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    do you mean $acadbcb^{-1}d$?2011-05-08
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    First, I do not understand exactly why all the vertices of $P$ are mapped to the same point of the quotient space. The argument in Munkres is kind of vague to me at least. Can you help? EDIT: The same Beatty, only now I have registered.2011-05-10
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    EDIT 2: @Qwirk: and yes, the labelling scheme is actually $acadbcb^{-1}d$. Sorry2011-05-10

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