  In the 1970s and again in the 1990s, Gromov gave a number of theorems and
conjectures motivated by the notion that the real homotopy theory of compact
manifolds and simplicial complexes influences the geometry of maps between
them. The main technical result of this paper supports this intuition: we show
that maps of differential algebras are closely shadowed, in a technical sense,
by maps between the corresponding spaces. As a concrete application, we prove
the following conjecture of Gromov: if $X$ and $Y$ are finite complexes with
$Y$ simply connected, then there are constants $C(X,Y)$ and $p(X,Y)$ such that
any two homotopic $L$-Lipschitz maps have a $C(L+1)^p$-Lipschitz homotopy (and
if one of the maps is a constant, $p$ can be taken to be $2$.) We hope that it
will lead more generally to a better understanding of the space of maps from
$X$ to $Y$ in this setting.
