For a given null-cobordant Riemannian
n
n
-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on
n
n
. In the appendix the bound is improved to one that is
O
(
L
1
+
ε
)
O(L^{1+\varepsilon })
for every
ε
>
0
\varepsilon >0
.
This construction relies on another of independent interest. Take
X
X
and
Y
Y
to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose
Y
Y
is simply connected and rationally homotopy equivalent to a product of Eilenberg–MacLane spaces, for example, any simply connected Lie group. Then two homotopic
L
L
-Lipschitz maps
f
,
g
:
X
→
Y
f,g:X \to Y
are homotopic via a
C
L
CL
-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces
Y
Y
.