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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.


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Could you expand a bit on what you have tried and what your thoughts are? – t.b. May 7 '11 at 20:29
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 $\mathbb{Z}_2 \oplus \mathbb{Z}^3$? – Aaron Mazel-Gee May 8 '11 at 4:17
do you mean $acadbcb^{-1}d$? – Juan S May 8 '11 at 10:50
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. – TJIF May 10 '11 at 23:11
EDIT 2: @Qwirk: and yes, the labelling scheme is actually $acadbcb^{-1}d$. Sorry – TJIF May 10 '11 at 23:32

As others have stated, it is best to show where you are confused.

From this section in Munkres, the first homology group is calculated as the abelization of the fundamental group. So firstly you need to calculate the fundamental group of the space. There are some useful theorems (and examples!) in Section 74 to help with this.

Now to get the homology group you have the relation

$$H_1(X) = \Pi_1(X,x_0)/[\Pi_1(x,X_0),\Pi_1(x,X_0)]$$

(Quotient by the commutator subgroup).

A useful theorem is Theorem 75.1

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