Homotopic chain maps have equal extended d-homology

The homotopy of chain maps on pre-abelian categories is investigated and the equality of extended d-homologies of homotopic chain maps is established. Singular and simplicial extended d-homology are introduced. If X and Y are of the same homotopy type, then their n-th singular extended d-homology R-modules are isomorphic. As a special case, if X is a contractible space, then its n-th singular extended d-homology R-modules for ≭ 0 are trivial