The Relation between Compact and Non-compact Equivariant Cobordisms

We show that the theory of stable complex G-cobordisms, for a torus G, is embedded into the theory of stable complex G-cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex G-cobordism theory does not lead to new relations. 1. Proper abstract moment maps. The main objective of this paper is to show that geometric G-equivariant cobordism theory is embedded in a similar theory for non-compact manifolds. Recall that two compact G-manifolds, where G is a compact Lie group, are said to be cobordant if their disjoint union is a boundary of a compact G-manifold. Often the manifolds are assumed to carry an additional structure preserved by the action and this structure is assumed to extend over the cobordism. The structures important for our present purpose are the orientation or/and tangential stable complex structure. (For definitions, see Appendix A of this paper or Chapter 28 by G. Comezaña in [Ma].) The cobordism classes of G-manifolds are then referred to as (geometric) oriented or stable complex G-cobordisms. We restrict our attention to the case where G is a torus. (The adjective “geometric”, omitted from now on, is used here to distinguish the cobordism theory we consider from its homotopy theoretic counterpart. See, e.g., Chapter 15 by S. R. Costenoble in [Ma].) In [Ka] we introduced a cobordism theory whose objects are non-compact G-manifolds with yet an additional structure called a proper abstract moment map. (We will recall its definition below.) The reason for considering non-compact manifolds is that this allows one to obtain a simple form of the linearization theorem (see [GGK1] and [Ka]). In its non-compact version proved in [Ka], the linearization theorem claims that under certain natural hypotheses every G-manifold is cobordant to the normal bundle to the fixed point set with a suitable proper abstract moment map. The addition of noncompact manifolds to a cobordism theory could, however, create a problem. Date: September 29, 1998. The authors are supported in part by the NSF and by the BSF. To appear in Proceedings of the International Workshop on Topology, M. Farber, W. Lueck, and S. Weinberger (Editors); Publ. AMS, Contemporary Math. Series.