Let
M
n
{M^n}
and
N
n
{N^n}
be closed manifolds, and let
G
G
be any (nonzero) module. (1) If
f
:
M
3
→
N
3
f:{M^3} \to {N^3}
is
C
3
{C^3}
G
G
-acyclic, then there is a closed
C
3
{C^3}
3
3
-manifold
K
3
{K^3}
such that
N
3
#
K
3
{N^3}\# {K^3}
is diffeomorphic to
M
3
{M^3}
, and
f
−
1
(
y
)
{f^{ - 1}}(y)
is cellular for all but at most
r
r
points
y
∈
N
3
y \in {N^3}
, where
r
r
is the number of nontrivial
G
G
-cohomology
3
3
-spheres in the prime decomposition of
K
3
{K^3}
. (2) If
f
:
M
3
→
M
3
f:{M^3} \to {M^3}
or
f
:
S
3
→
M
3
f:{S^3} \to {M^3}
is
G
G
-acyclic, then
f
f
is cellular. In case
G
G
is
Z
Z
or
Z
p
{Z_p}
(
p
p
prime), results analogous to (1) and (2) in the topological category have been proved by Alden Wright. (3) If
f
:
M
n
→
M
n
f:{M^n} \to {M^n}
or
f
:
S
n
→
M
n
f:{S^n} \to {M^n}
is real analytic monotone onto, then
f
f
is a homeomorphism.