Final answer:
The question involves computing homology groups for a closed orientable surface of genus two and understanding the resulting topological spaces when circles on the surface are collapsed. Homology groups relate to the topology of the surface, and quotient spaces alter the topology when circles are collapsed to points.
Step-by-step explanation:
The question asks to compute the homology groups Hn(X, A) and Hn(X, B) for a closed orientable surface X of genus two with A and B being the given circles. Additionally, it inquires about the spaces X/A and X/B.
For an orientable surface of genus two, the homology groups can generally be found using the relation between the genus and the Betti numbers. The homology group H1 for such a surface has rank 4 as there are two 'handles,' each contributing two generators to H1. When factoring out a circle like A or B, you effectively 'pinch' the surface at that circle, affecting the homology by effectively eliminating a generator associated with that circle.
As for X/A and X/B, these are quotient spaces formed by collapsing the respective circles to a point. These operations can change the topology of the surface and need to be considered when calculating homology groups.