  This paper provides a convenient and practical method to compute the homology
and intersection pairing of a branched double cover of the 4-ball.
  To projections of links in the 3-ball, and to projections of surfaces in the
4-ball into the boundary sphere, we associate a sequence of homology groups,
called the disoriented homology. We show that the disoriented homology is
isomorphic to the homology of the double branched cover of the link or surface.
We define a pairing on the first disoriented homology group of a surface and
show that this is equal to the intersection pairing of the branched cover.
These results generalize work of Gordon and Litherland, for embedded surfaces
in the 3-sphere, to arbitrary surfaces in the 4-ball. We also give a
generalization of the signature formula of Gordon-Litherland to the general
setting.
  Our results are underpinned by a theorem describing a handle decomposition of
the branched double cover of a codimension-2 submanifold in the $n$-ball, which
generalizes previous results of Akbulut-Kirby and others.
