Calculating $\pi_2(X\cup_\alpha e_\alpha)$ using Hurewics theorem and covering spaces Consider the CW-complex $X$ obtained by wedging two circles. Denote by $a$ and $b$ the generators of $\pi_1(X)$. On $X$, attach two discs with attaching maps
\begin{align*}S^1\stackrel{a^5(ab)^{-2}}{\longrightarrow}X\quad\quad S^1\stackrel{b^3(ab)^{-2}}{\longrightarrow}X\end{align*}
Call the resulting CW-complex $W$. I want to calculate $\pi_2(W)$. My idea is as follows: Let $\widetilde{W}\stackrel{p}{\longrightarrow}W$ be the universal covering space of $W$. In dimensions 2 or higher, $p_*$ is an isomorphism so by Hurewicz Theorem we obtain $H_2(\widetilde{W})\cong \pi_2(\widetilde{W})\cong\pi_2(W)$. Having a good understanding of the CW-complex structure on $\widetilde{W}$ could then potentially solve the problem, but this is where I get stuck: Since $X\hookrightarrow W$ does not induce isomorphism in $\pi_1$, we do not get from the (general) theory, see eg. Lemma 4.38+proof in Hatcher, that $\widetilde{W}$ is obtained by lifting the cells of $W$ via the covering of $X$. I know that the covering space of $X$ is the Cayley graph, and I have been considering how to appropriately attach 2-cells to obtain a universal cover of $W$, but in vain.
My question is, can someone tell me how to obtain a universal cover of $W$? Otherwise, if the strategy appears to be a dead end, I would appreciate hints to proceed in a more fruitful direction.
Thanks in advance.
Edit: For an alternative reference for finding an explicit CW-structure on a universal cover in a situation such as above, see Hatcher section 1.3 "Cayley Complexes".
 A: The process you are looking for is called free Fox differentiation.  See for example page 3 of this book.
First fix a lift of the basepoint $*\in \widetilde{W}$.
Each directed $1$-cell $e_i$ in $W$ lifts to a unique $1$-cell  $\widetilde{e_i}$ of $\widetilde{W}$, rooted at $*$.  In general, the $1$-cells of  $\widetilde{W}$ are the translates of these.  That is $\widetilde{e_i}g$, for $g\in \pi_1(W)$.
The $1$-cell $\widetilde{e_i}g$ goes from $*g$ to $*g_ig$, where $e_i$ represents $g_i\in \pi_1(W)$.
For each 2-cell of $W$, with boundary a word in the $g_i$ (and their inverses), we can draw a lift in $\widetilde{W}$ as a polygon with edges translates of the directed $1$-cells in the word.    If we set the first vertex to represent $*$, what we must translate the remaining vertices and edges by is determined inductively.
The coefficient on $\widetilde{e_i}$ of the boundary of a $2$-cell with boundary word $w_j$ is then the free Fox derivative $\frac{\partial w_j}{\partial g_i}$.
In general, for a word $w$ in symbols $g_i^{\pm1}$, representing elements of a group $G$,the symbol $\frac{\partial w_j}{\partial g_i}\in \mathbb{Z}[G]$ denotes the sum of group elements represented by the word to the right of an instance of $g_i$ in $w_j$, minus group elements represented by the word to the right of an instance of $g_i^{-1}$ in $w_j$, starting from the $g_i^{-1}$.
Note, if the group action on the cover is a right action, then we must read words right to left, when interpreting as paths.
Thus $\pi_2(W)$ for a $2$-complex $W$ may computed algebraically as the kernel of the map over $\mathbb{Z}[\pi_1(W)]$, represented by the matrix with entries $\frac{\partial w_j}{\partial g_i}$.
There is a similar purely algebraic calculation for $\pi_3(W)$, when $W$ is a $2$-complex - see for example this paper.
