Imbeddings and homology cobordisms of lens spaces

Abstract

In this paper we consider the existence of smooth or PL imbeddings of manifolds in Euclidean space with codimension one. The manifolds we treat are made from lens spaces (or homotopy lens spaces) by removing a disc or by taking a connected sum. (It is easy to see [R2] that a homotopy lens space must be punctured in order to imbed in Euclidean space of one higher dimension.) The results of [GL, R2] show that this problem reduces to the problem of finding a homology cobordism (i.e. one with the homology of a product) between two homotopy lens spaces. It is shown in [R2] that for (linear) lens spaces L with :~(L) of prime power order, the existence of such a homology cobordism implies the existence of an s-cobordism, and hence that a lens space L imbeds punctured if and only if L admits an automorphism satisfying certain conditions. It is straightforward to explicitly describe all such lens spaces. Further, the connected sum of two such lens spaces imbeds if and only if they are diffeomorphic. Hence in both problems, the homology cobordism may be taken to be a product. The present paper will demonstrate that the situation changes when the order of nl(L) is divisible by more than one prime and when L is allowed to be a homotopy lens space. The invariants used in [R2] as obstructions to imbedding were equivariant signatures associated to coverings of prime-power degree; in the general case considered here they do not characterise a homotopy lens space, even up to h-cobordism. Nevertheless, we show that in dimensions greater than three, the signature invariants used in JR2] do determine a homotopy lens space up to homology cobordism within its normal cobordism class. Hence only a small portion of the invariants used in [W1] to classify homotopy lens spaces comes into play; in particular Reidemeister torsion plays no role. This classification up to homology cobordism leads to necessary and sufficient conditions for punctured imbeddings and imbeddings of connected sums. The fact that only the invariants associated to prime-power coverings come into play has an analog in other parts of topology, most notably in the theory of transformation groups. In that context, Smith theory [B1] provides restrictions on