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This paper determines the
surgery obstructions for all surgery problems of the form
as explicit elements in the surgery obstruction groups Ln+2h where σ : K4k+2 → S4k+2
is the usual Kervaire problem and Mn is a closed, compact, oriented manifold with
π1(M) finite. Due to the well known observation that the surgery obstruction for a
surgery problem on a closed manifold depends only on the resulting cobordism class
in Ω∗(Bπ1(M) × G∕TOP), this is the fundamental step in obtaining the surgery
obstructions for all surgery problems over closed manifolds, as long as π1(M) is finite.
(In the case π1(M) infinite, the situation is much more complex. A key part of
the question would be resolved if one could prove the Novikov conjecture
though.)
One of our main results is that only three types of obstruction can occur. This is,
in fact, the first step in proving the oozing conjecture. The proof is completed in part
II of this paper where we give characteristic class formulae for evaluating these
obstructions.
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