I wish to put a CW complex structure on a space $X$ obtained as follows. Delete the interiors of two disjoint subdiscs in the interior of $D^2$, and then identify all three resulting boundary circles via homeomorphisms preserving clockwise orientation of these circles.
I know how to put a CW complex structure on the space before the resulting boundaries are identified. There are 3 0-cells, 5 1-cells and 1 2-cell, see my crudely drawn picture below to see how they're identified. But I don't really know how to go from this to the actual desired space.
I guess this is the exercise in Hatcher's book. Consider the following CW structure: one $0$-cell $x$, three $1$-cells $a$,$b$ and $c$ and one $2$-cell $U$, whose boundary is given by $aba^{-1}b^{-1}ca^{-1}c^{-1}$.