A NEW PROOF OF THE TRIVIAL RANGE COMPLEMENT THEOREM

In this note we give a new proof of the following theorem. Theorem. Suppose Mn is a compact PL n-manifold and X and Y are compact subsets of the interior of Mn which have fundamental dimension less than or equal to k, 2k + 2 ≤ n, and which satisfy the inessential loops condition. If M −X and M − Y are homeomorphic, then X and Y have the same shape. For n ≥ 5, this is one half of a theorem which was first stated in [4]. The proof given here is simpler than the previous proofs of the theorem and also improves on those proofs in that it applies in all dimensions; in particular, it is valid when n = 4. The reason that the standard proof for the theorem is only valid for n ≥ 5 is that it makes use of the converse of the theorem (same shape implies homeo-morphic complements) and the converse is still not known to hold in dimension 4. In the proof given here, we depart from the standard pattern (which dates back to Chapman’s original proof of the complement theorem) of begining the proof by re-embedding the compacta in a hyperplane. The reader is referred to [2] for an explanation of the standard proof as well as for a complete history of the finit