文档介绍：LECTLTHES ON TEE h-COBORDISM THEOREM
JOHN MILNOR
L. SIEBEMvIANN
AND
J. somw
PRINCE'KIN, NEGJ JERSEY
PRINCETON UIVrVERSITY PRESS
1965
Copyright 0x965, by Princeton University Press
All Rights Reserved
Printed in the United States of America
Zhese are notes for lectures of John Milnor that were given
as a seminar on differential topology in October and November,
1963 at Princeton University.
Let W be pact smooth manifold having two boundary
components V and V1 such that V and V' are both deform-
ation retracts of W. Ben W is said to be a h-cobordism
between V and Vt . The h-cobordism theorem atates that if in
addition V and (hence) V1 are simply connected and of dimen-
sion greater than 4 , then W is diffeomorphic to V X [O, 11
and (consequently) V is diffeomorphic to V' . Ihe proof is
due to Stephen Smale [6]. 'Ihis theorem has numerous important
applications - including the proof of the generalized ~oincare
conjecture in dimensions > 4 - and eeveraJ. of these appear
in $9. Our ma,in task, however, is to describe in some detail a
proof of the theorem.
Here is a very rough outline of the proof. We begin by
constructing a Morse function for W (), . a smooth
function f : w -> [o, 11 with v = C1(0) , v1 = f1(1)
such that f has finitely many critical points, all nondegen-
erate and in the interior of W. 'ihe proof is inspired by the
observation () that W is diffeomorphic to V X [o, 11 if
(and only if) W admits a brae f'unction as above with no crit-
ical points. %us in 554-8 we show that under the hypothesis
of the theorem it is possible to simplify a given Morse function
f until finally all critical points are eliminated. In $4, f
is ad jueted so that the level f (p) of a critical point p is
an increasing f'unction of its index. In $5, geometrical condi-
tions are given under which a pair of critical points p, q of
index h and h + 1 can be eliminated or *cancelleds. In