This paper provides a convenient and practical method for calculating the homology and cross-point pairing of 4-ball bifurcated double covers.
We associate a sequence of homology groups, called disoriented homology, with the projection of the links in the 3-ball and the projection of the surface in the 4-ball onto the bounding sphere. Disoriented homology indicates that it is isomorphic to the double-branched cover homology of a link or surface. We define the pairing at the first non-directed homology group on the surface and show that this is equivalent to the cross-pairing of the divergent cover. These results generalize the work of Gordon and Litherland to apply surfaces embedded in a 3-sphere to arbitrary surfaces in his 4-sphere. We also generalize the Gordon-Litherland signature equation to a general setting.
Our results are underpinned by a theorem that describes the handle decomposition of bifurcated double covers of codimensional 2-submanifolds of $n$ balls, generalizing previous results by Akbulut-Kirby and others.