1960/09/01 by C. T. C. Wall · 4 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Mathematics #Cobordism #Ring (chemistry) #Pure mathematics #Algebra over a field
paper · doi:10.2307/1970136
openalex publication_date 1960/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/04/05
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