We determine characteristic
class formulae for surgery problems over any compact oriented and closed manifold
with finite fundamental group. This essentially evaluates the boundary maps in the
surgery exact sequences
(The final step in determining ∂ is carried out in a sequel which represents joint work
with I. Hambleton, L. Taylor, and B. Williams.) These formulae involve the L-genus
of M, pullbacks of the classes K4i and k4i+2 in H∗(G∕TOP) and characteristic
classes of the universal covering of M (coming from H∗(Bπ1(M))). It turns out that
only classes in the first three of these groups, ∗ = 1,2,3, are needed. This can be
interpreted as saying that only codimension 1, 2, and 3 submanifolds are
needed to determine the surgery obstruction. In this form our result was
originally conjectured twenty years ago and has become known as the oozing
conjecture.
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