Cobordism

We have already defined the word ‘cobordism’ in §5.1: recall that if W is a manifold, and ∂ - W and ∂ + W are disjoint manifolds with union ∂W , we call the pair ( W, ∂ - W ) a cobordism and the pair ( W, ∂ + W ) the dual cobordism; and also call W a cobordism of ∂ - W to ∂ + W and say that ∂ - W , ∂ + W are cobordant. In the earlier chapter, we were concerned with the geometry of a particular cobordism. We now observe that being cobordant is an equivalence relation amongst diffeomorphism classes of manifolds. For M × I is a cobordism of M to itself; if W is a cobordism from M 0 to M 1 then the same manifold, but with ∂ ± W interchanged, is a cobordism from M 1 to M 0 ; and if W 0 is a cobordism from M 0 to M 1 and W 1 is a cobordism from M 1 to M 2 , then glueing W 0 to W 1 along M 1 gives a cobordism from M 0 to M 2 . For this relation not to be vacuous, we insist throughout that the manifolds W in question be compact: otherwise the product M × [0 , 1) would give a cobordism of any manifold M to the empty set. The simple definition just given already leads to interesting results, but the concept of cobordism lends itself to a wide variety of possible generalisations and restrictions, and these lead to a flexible tool in the study of manifolds. For example, we may choose to restrict the manifolds ( and cobordisms) to be oriented, weakly complex, or k -connected (for a fixed k ); we may add the structure of a map to a fixed space X ; if X is a manifold, we may further require this map to be an embedding, or an immersion. We may consider pairs ( M,V ) with V a submanifold of M and then cobordisms ( N,W ) with W a submanifold of N (and ∂ - W = V , ∂ - N = M ), where wemay also fix the group of the normal bundle. Next we consider pairs, where M is a manifold and defines a smooth action of the compact Lie group G on M .We may also restrict the orbit types of the action to lie in an assigned closed set of orbit types - an extreme example is the class of fixed-point-free actions.