Is $<a,b^2>$ isomorphic to $\mathbb{Z}^3$?

Question

I have this question from abelianizing $<a,b^2 , bab^{-1}>$, trying to compute the first singular homology of a covering space of the figure eight space.

Answer 1

From your comment, I understand that you are trying to abelianize the fundamental group of a certain double covering space $X$ of the figure 8 (that is depicted in Hatcher's textbook as you say in your comment) as follows. That graph $X$ has two vertices $V,W$ and four edges each of which is labelled according to its image in the figure 8 under the covering map: $X$ has one edge labelled $a$ going from $V$ to $V$; and one edge labelled $a$ going from $W$ to $W$; and one edge labelled $b$ going from $V$ to $W$; and one edge labelled $b$ going from $W$ to $V$.

But, although these labels $a,b$ represent generators of the fundamental group of the figure 8, they do not represent generators of $\pi_1(X)$. You cannot manipulate them in a presentation of the fundamental group of $\pi_1(X)$ as if they were generators. Thus, your comment "if I abelianize it, $bab^{-1}$ becomes $a$" does not apply, because although your statement is true in the figure 8, it is false in $X$.

So, it is better to label the edges of $X$ with unique labels, so that you can tell them apart. A good way to do this is to use different subscripts on edges of $X$ which map to the same edge of the figure 8. For instance, you can use $a_1$ for the edge of $X$ from $V$ to $V$; $a_2$ for the edge from $W$ to $W$; $b_1$ for the edge from $V$ to $W$; and $b_2$ for the edge from $W$ to $V$.

Now you must follow the formula for determining generators of the fundamental group of a graph: choose a base point in $X$, let's say $V$; choose a maximal tree in $X$, let's say the edge $b_1$; and then for each of the three edges $e \in \{a_1,a_2,b_2\}$ not in the maximal tree write down a closed curve based at $V$ that passes over $e$ and not over either of the other two. Also, as you write down those closed curves, give new names to each.

So, the closed curve that passes over $a_1$ is $c = a_1$; the closed curve that passes over $b_2$ is $d = b_1 b_2$; and the closed curve that passes over $a_2$ is $e = b_1 a_2 b_1^{-1}$.

Now you can make your conclusions: the fundamental group of $X$ is the free group with basis $c,d,e$; and its abelianization is the free abelian group with basis $c,d,e$.