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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
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