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 M0 to M1 then the same manifold, but with ∂±W interchanged, is a cobordism from M1 to M0; and if W0 is a cobordism from M0 to M1 and W1 is a cobordism from M1 to M2, then glueing W0 to W1 along M1 gives a cobordism from M0 to M2. For this relation not to be vacuous, we insist throughout that the manifolds W in question be compact: otherwise the productM × [0, 1) would give a cobordism of any manifoldM 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) withW 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.