We prove a global implicit function theorem. In particular we show that any Lipschitz map f:ⁿ× ^m→ⁿ (with n-dim. image) can be precomposed with a bi-Lipschitz map g:ⁿ× ^m→ ⁿ× ^m such that f∘gˉ will satisfy, when we restrict to a large portion of the domain E⊂ ⁿ× ^m, that f∘gˉ
Nearby in the stack
is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map
gˉ
distorts
ⁿ+m
in a controlled manner, so that the fibers of
f
are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set
E
on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman's 1979 construction of a
C1
map from
[0,1]3
onto
[0,1]2
with rank
≤1
everywhere. On route we prove an extension theorem which is of independent interest. We show that for any
D≥n
, any Lipschitz function
f:[0,1]ⁿ→ ^D
gives rise to a large (in an appropriate sense) subset
E⊂[0,1]n
such that
f∣E
is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of
ⁿ
. The most interesting case is the case
D=n
. As a simple corollary, we show that
n
-dimensional Ahlfors-David regular spaces lying in
^D
having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in