I am trying to find all connected covers of the following space $X$ (up to isomorphisms)
$X$ has one $0$-cell, two $1$-cells labeled $a$ and $b$, and three $2$-cells attached via $a^2$, $b^2$ and $aba^{-1}b ^{-1}$ respectively.
So far I have computed the fundamental group (the Klein-four group generated by $a$ and $b$). Since it has $3$ proper subgroups I wanted to find the corresponding covers, but had no luck so far..
Could someone help me with how to start?
On
One question is how to represent the cover! It is best first to use the $1$-dimensional part. So here on the left
is a picture of the $1$-dimensional part of the universal cover, taken from p. 590 of Nonabelian Algebraic Topology, in a section on covering morphisms of groupoids. The picture on the right is of the boundary of one of the cells covering your $2$-cell. An edge labelled of the form $(a, -)$ maps down to $a$.
Hope this enables you to draw the other pictures, and other examples.
It may be helpful to think of $X$ as being the 2-skeleton of a suitable cell decomposition of the 4-manifold $\mathbb{RP}^2\times\mathbb{RP}^2$. Then the universal cover will be a subcomplex of $S^2\times S^2$, and the other covers of $X$ can be viewed as subcomplexes of suitable quotients of the manifold $S^2\times S^2$.