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8.7.3. Theorem.
(The h-cobordism Theorem) Let
W
n
be a compact smooth n-
dimensional manifold which is a cobordism between submanifolds
V
and
V
¢ satisfy-
ing the following conditions:
(1) All three spaces
W
,
V
, and
V
¢ are simply connected.
(2) The inclusion i :
V
Ã
W
induces isomorphisms i
q*
:
H
q
(
V
) Æ
H
q
(
W
) for all q.
(3) n ≥ 6.
Then
W
is diffeomorphic to
V
¥ [0,1].
Proof.
See [Miln65a].
Definition.
Let
W
n
be a compact smooth n-dimensional manifold and which is a
cobordism between submanifolds
V
and
V
¢. If both
V
and
V
¢ are deformation retracts
of
W
, then
W
is said to be an
h-cobordism
between
V
and
V
¢ and
V
and
V
¢ are said
to be
h-cobordant
.
Theorem 8.7.3 gets its name from the fact that the hypotheses on
W
,
V
, and
V
¢
made
W
into an h-cobordism. (One needs Theorem 7.4.3.7 to see this.) There are some
important corollaries.
8.7.4. Corollary.
Two simply connected closed smooth n-dimensional manifolds,
n ≥ 5, which are h-cobordant are diffeomorphic.
The next corollary provides a characterization of the n-disk.
8.7.5. Corollary.
Let
W
n
be a compact smooth simply connected n-dimensional
manifold, n ≥ 6, with simply connected boundary. Then the following four assertions
are equivalent:
(1)
W
n
is diffeomorphic to
D
n
.
(2)
W
n
is homeomorphic to
D
n
.
(3)
W
n
is contractible.
(4)
W
n
has the same homology groups as a point.
In 1904 Poincaré conjectured the following:
The Poincaré Conjecture:
Every compact simply connected closed three-
dimensional manifold is homeomorphic to
S
3
.
A generalization of this conjecture can be proved.
8.7.6. Corollary.
(The Generalized Poincaré Conjecture) If
M
n
, n ≥ 4, is a closed
compact simply connected smooth manifold which has the same homology groups as
the n-sphere
S
n
, then
M
n
is homeomorphic to
S
n
.
Proof.
When n ≥ 5, then the result is an easy consequence of Corollary 8.7.5 and
Theorem 7.5.2.7. The case n = 4 is more difficult and was proved in [Free82]. Note