Determination of the Cobordism Ring

The cobordism ring was first defined by R. Thom 15, and is sometimes known as the Thom algebra. Consider the set of closed oriented manifolds of dimension k (here, and throughout this paper, all manifolds are supposed differentiable, of what class it does not matter), and if V is an oriented manifold, denote by V the same manifold with the opposite orientation. Introduce the relation VW (pronounced: V is cobordant with W) if there is a compact oriented manifold M with oriented boundary 8,(M) = V + (W), where + denotes disjoint union. It is easy to see that is an equivalence relation, compatible with + and -, so that the equivalence classes form an abelian group, f2,, the cobordism group in dimension k. Since, if V is closed, &0(M x V) = &0M x V, topological product is compatible with -, and induces a product i hence the name 'intrinsic homology' adopted by Rohlin for cobordism. If orientation is not required in the above, we obtain an equivalence relation V -2 W (pronounced: V is cobordant with W mod 2) for nonoriented manifolds, and a new cobordism ring St = Ek Sk We will denote by r : i2 9f the natural map obtained by ignoring orientation. Q2, T are rings in the ordinary algebraic sense, and r is a homomorphism between them, and the problem with which we are concerned is to give a purely algebraic description of them. Now the structure of T was already completely determined in 15: 9 is a ring of polynomials mod 2, with one generator x, in each dimension i not of the form 2' 1. A necessary and sufficient condition that two manifolds be cobordant mod 2 is that they have the same Stiefel numbers, which are defined as follows. Let wI be the ith Stiefel class mod 2 of the manifold Mk, so wI e HI(Mk, Z2). (For a definition of the Stiefel classes of a manifold see 5 or 13.) Form any homogeneous polynomial of degree k in the wf, f(w, ***.., w) e H(Mk, Z2) and evaluate it on the fundamental cycle (mod 2) of Mk. It is frequently convenient to regard the wI as the elementary symmetric functions of k (or even more) inde292