The bordism groups

Abstract

Given a pair (X, A) of spaces, consider all maps f:(B _n , $\dot B^n$) → (X, A), where B _n is a compact oriented differentiable manifold with boundary $\dot B^n$. We introduce a relation of the bordism type on the class of all such f, arriving in § 4 at the set Ω _n (X, A) of equivalence classes. These groups constitute a generalized homology theory; it is shown in § 5 that the Eilenberg-Steenrod axioms are satisfied except for the dimensional axiom. In § 12 we give a homotopy interpretation for Ω _n (X, A). It is shown that Ω _n (X, x ₀), X a CW complex with base point x ₀∈ X, is given by the homotopy group π _n+k (X ∧ MSO (k)); here MSO (k) is the Thorn space and k ≧ n + 2. In § 13 the dual generalized cohomology theory is sketched; this is due to Atiyah [1]. We try to fill in along the way some of the background material of differential topology.