Bordism theories and chain complexes

Abstract

Although every generalized homology theory h ∗ is related to a chain functor C ∗ , some homology theories (like e.g. bordism theories) admit a much simpler chain theoretical description by means of chain theories with models. Algebraically these models are playing the role of relative cycles of a specific chain functor, while geometrically they behave like singular manifolds in bordism theory. A necessary and sufficient condition is formulated (using chain functors) ensuring that a homology theory is associated with a chain theory with models.