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 $n\neq 0$ are trivial .