6
CHARLES CONLE Y
type o f th e pointe d spac e obtaine d fro m a bloc k o n collapsin g th e se t o f exi t point s t o
one point .
3.3. The index of the rest points. Fo r example , the middl e res t poin t in the botto m
portrait o f Figur e 2 admits a block in the for m o f a n interval bot h o f whos e en d points ar e
exit points . Th e space obtaine d o n collapsing these en d points to on e point i s a circle
(Figure 3) . Thu s the index o f thi s res t poin t i s a pointed circle , or one-spher e (o r mor e
precisely, its homotopy type) .
•-
j +
o
FIGURE 3
The rest point i n th e to p portrai t admit s a block in th e for m o f a n interval wit h
empty exi t set . Whe n th e empt y se t o f a space is collapsed t o a point th e resultin g space is
homeomorphic t o th e disjoin t unio n o f th e give n space an d th e (distinguished ) point . I n
this cas e it i s the disjoin t unio n o f th e interva l an d th e point . Thi s spac e therefor e i s a
representative o f the homotopy typ e whic h is the index . A simpler representativ e i s ob-
tained b y deformin g th e interva l t o a point. Th e resulting (pointed) tw o poin t spac e is
called th e pointe d zero-sphere .
Now since the zero-spher e is not homotopicall y equivalen t t o th e one-sphere , the in-
variance o f th e index unde r continuatio n implie s that th e res t poin t i n th e to p portrait can -
not b e continued t o th e middl e on e in th e botto m portrai t n o matter how large a family o f
equations i s taken. S o at least som e equivalence classe s ar e distinguishe d b y th e index . Th e
writer ha s not decide d whethe r o r not the y al l are.
All of th e res t point s i n th e botto m portrai t ar e nondegenerate critica l point s o f a
gradient flow : f(x) = x(l - x
2)
- X is, of course , the gradien t o f a function, namel y
x2/2
- x
4/4
- X x Therefor e eac h o f thes e point s has a classical Mors e index whic h is, by
definition, th e numbe r o f positiv e eigenvalue s o f th e linearizatio n o f the equatio n a t th e
rest poin t (i n othe r words , the dimensio n o f th e unstabl e manifold) . Therefor e th e inde x
of th e right - and left-hand res t point s in th e botto m portrai t o f Figur e 2 is zero whil e tha t
of the middl e poin t i s one. In genera l a nondegenerate res t poin t o f a gradient flo w is iso-
lated a s an invariant se t an d the inde x here define d i s a pointed spher e whos e dimensio n i s
the classica l Mors e index.
The right-han d res t poin t i n th e middl e portrai t i s degenerat e an d th e classica l Mors e
index i s not define d fo r thi s point . However , the poin t i s an isolated invarian t se t an d a
block ca n b e chosen i n th e for m o f a n interva l on e o f whos e en d point s i s an exit point .
The index is , therefore, th e homotopy typ e o f a pointed interval . A simpler representativ e
is obtained o n deformin g th e interva l t o th e poin t —thus the index i s the homotopy typ e
of the (pointed ) on e poin t space .
Using the interva l indicate d i n th e middl e portrai t o f Figur e 2 and increasin g X , it is
seen tha t thi s res t poin t continue s t o th e empt y set . Th e empty se t i s always a n isolate d
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