Pseudohomology and homology

The notion of a pseudocycle is introduced by McDuff and Salamon (J-holomorphic curves and quantum cohomology, University Lecture Series, Vol. 6, AMS (1994)) to provide a framework for defining Gromov-Witten invariants and quantum cohomology. This paper studies the bordism groups of pseudocycles, called pseudohomology groups. These are shown to satisfy the Eilenberg-Steenrod axioms. For smooth compact manifold pairs, it is proved that pseudohomology is naturally equivalent to homology. Easy examples show that this equivalence does not extend to the case of non-compact manifolds.