A continuous surjection π :
X → Y between topological spaces is called “ductile” if, for each y ∈ Y and
neighborhood U of y there is a neighborhood V of y which contracts to y through U
in such a way that this contraction can be covered by a homotopy of π−1(V ). It is
shown, in this note, that if π : X → Y is ductile and Y is paracompact then the
inclusion of the image π∗C∗(X) of the singular chain complex of X in the singular
chain complex C∗(Y ) of Y induces an isomorphism in homology. Thus H∗(Y ) can be
computed from those singular simplices of Y which are images of singular simplices of
X.
