Vol. 151, No. 1, 1991

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Surgery with finite fundamental group. II: The oozing conjecture

R. James Milgram

Vol. 151 (1991), No. 1, 117–150
Abstract

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

⋅⋅⋅→∂Lhn+1(Z π) → ℋ𝒯 (M n) → [M n,G∕T OP ] ∂→ Lhn(Z π).

(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.

Mathematical Subject Classification 2000
Primary: 57R67
Secondary: 18F25, 19J25
Milestones
Received: 23 March 1988
Revised: 25 May 1989
Published: 1 November 1991
Authors
R. James Milgram
Stanford University
United States