I am asked to classify the compact surface obtained by pasting the edges of a polygonal region with the labeling scheme $abcdabdc$ and compute it's first homology group. I classify the space as the connected sum of 3 real projective planes using cutting and pasting. (A quick algebraic check: call our scheme $w$ and let $e = abd$. Then $w = abcb^{-1}a^{-1}e^2c$. Let $f = b^{-1}a^{-1}e^2c$, then $w = abe^{-2}abf^2$. Let $g = e^{-2}ab$, then $w = e^2g^2f^2$ so our space can be classified as the connected sum of 3 projective planes).
The first homology group of the $n$-fold connected sum of real projective planes is known to be $\mathbb{Z}^{n-1} \oplus \mathbb{Z}_2$ (see here: Homology of connected sum of real projective spaces) so I deduce $H_1(X) \cong \mathbb{Z}^2 \oplus \mathbb{Z}_2$.
Now consider $H_1(X)$ as the abelianization of the fundamental group $\pi_1(X) \cong \langle a, b, c, d | abcdabdc = 1\rangle$. Then $H_1(X) \cong \text{Ab}(\pi_1(X)) \cong \langle a, b, c, d |2(a+b+c+d)=0 \rangle$. This representation is justified here: Presentation of the abelianization of $G$. If we consider the generators of this presentation to be $\{a, b, c, a+b+c+d\}$ it seems clear that $H_1(X) \cong \mathbb{Z}^3 \oplus \mathbb{Z}_2$. (If you're not convinced that this manipulation on the generators is correct it is also shown here Presentation of abelian group that we can use the Smith Normal Form which gives the same result).
Clearly $\mathbb{Z}^2 \oplus \mathbb{Z}_2 \ne \mathbb{Z}^3 \oplus \mathbb{Z}_2$ so what went wrong?
Best Answer
If you identify your edges of an octagon in the pattern $abcdabdc$ then its eight vertices are identified into two points $P$ and $Q$ say, where $Q$ is the image of the vertices between the $a$ and $b$ edges and $P$ is the images of the other six vertices. Therefore the edges $a$ and $b$ are not mapped into loops in the surface, but $c$ and $d$ are mapped to loops based at $P$. If we replace the $ab$ pairs of edges with edges labelled $e$, say, then $e$ becomes a loop in the surface, and the fundamental group is $\langle c,d,e\mid ecdedc=1\rangle$, Abelianising to $\Bbb Z_2 \oplus\Bbb Z^2$.