Chain theories and their derived homology
R. O. Burdick, Pierre E. Conner, E. E. Floyd · Proceedings of the American Mathematical Society · 1968
In this note we shall prove that the generalized homology theory known as oriented bordism [l] cannot be derived from an underlying chain theory.This will follow from our main theorem which characterizes those generalized homologies which arise from chain theories.Consider a covariant functor L from the category of finite CW-pairs to the category of chain complexes.For each finite CW-pair (X, A) we thus have the chain complex L(X, A): ■ ■ ■ *-Ln_x(X, A)*-Ln(X, A)*and for each map/: (X, A)->(F, B) we have a chain map L(f): L(X,A) ->L( F, B) of degree 0. We denote L(J) by ft; also L(X, 0) is denoted by L(X).We call L a chain theory if(1) for each finite CW-pair (X, A) the sequences