The homotopy group 7r' +1 (B, C) = 7rf+l(B, C, d), (n _ 1), of an arcwise connected' space B relative2 to an arcwise connected' subset C with base point d E C may be defined as the fundamental group of a certain space' an(B, C, d). The group rn+1 (B, C) is independent of the base point d in the sense that irf+l(B C, d,) and irf+l(B, C, d2) are isomorphic. Hurewicz demonstrates this by showing that 'n (B, C, di) and an(B, C, d2) belong to the same homotopy type. According to my generalization4 of Whitehead's theorem these function spaces are therefore deformation retracts of some containing space W. The containing space W constructed by this method is a subset of the function space a n(B, C, D), where D is the arc, with end-points di and d2 which appears in Hurewicz' (unpublished) proof. This suggests that a n(B, C, di) and an(B, C, d2) may be deformation retracts of an(B, C, D) itself. It will be shown below that this is indeed the case, at least if B and C are compact ANR-sets. (The reader will note that the reasoning can now be reversed to deduce the independence of 7rn +l(B, C) of the base point.) Furthermore deformation retraction of B or C induces deformation retraction of a7(B, C, d), hence leaves 7rf+l(B, C) unaltered. Before plunging into the proof I generalize the spaces 'an (B, C, D) by (a) generalizing the antecedent cells and cellboundaries to arbitrary topological spaces, and (b) removing the dependence of an on the obviously irrelevant number three. Thus we might loosely describe the investigation as a study of the relationship between deformation retraction in the image space and deformation retraction of the function space.