SURGERY WITH FINITE FUNDAMENTAL GROUP .1. THE OBSTRUCTIONS

This paper determines the surgery obstructions for all surgery problems of the form id x sigma: M2 x K4k+2 --> M(n) x S4k+2 as explicit elements in the surgery obstruction groups L(n+2)h where sigma: K4k+2 --> S4k+2 is the usual Kervaire problem and M(n) is a closed, compact, oriented manifold with pi-1(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 OMEGA-*(B-pi-1(M) x G/TOP), this is the fundamental step in obtaining the surgery obstructions for all surgery problems over closed manifolds, as long as pi-1(M) is finite. (In the case pi-1(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.