We prove the Gromov conjecture on the macroscopic dimension of the universal covering of a closed spin manifold with a positive scalar curvature under the following assumptions on the fundamental group.
0.1
0.1
. Theorem. Suppose that a discrete group
π
\pi
has the following properties:
1
1
. The Strong Novikov Conjecture holds for
π
\pi
.
2
2
. The natural map
p
e
r
:
k
o
n
(
B
π
)
→
K
O
n
(
B
π
)
per:ko_n(B\pi )\to KO_n(B\pi )
is injective. Then the Gromov Macroscopic Dimension Conjecture holds true for spin
n
n
-manifolds
M
M
with the fundamental groups
π
1
(
M
)
\pi _1(M)
that contain
π
\pi
as a finite index subgroup.