(7. i) j j i _~ j , ( f , , al) _ j , ( f o , ao)
in th'e invariants Ji in an admissible deformation f f depends only on the path (a t) and not on f
More explicit f o r m u l a s for 3:~ and d J i are needed. To t h a t end set
(7.2) d(h0 -- ui (rood I),
a n d recall t h a t ui can be t a k e n as ~ when ai is a zero, a n d o when ai is a pole.
L e t ei be i or - - I according as ai is a zero or a pole. T h e n
I I
(7.3) u i - - - - arg e , - (mod I).
2 ~ r 2
To make the f o r m u l a for J~ more definite we shall use a b r a n c h ~ X of t h e a r g u m e n t for which
o ~ a r g X < z ~
signalling this branch by the addition of the bar. I n accordance with the de- finition of C~ in (5-3)
C,(a,, ,) = B (a~, a) A'(~,_~) (a, -- ao) (i # o) B (ao, a),
Ci(ao, a) = A ' (ao, a) (a~ -- a,).
On r e f e r r i n g to t h e f o r m u l a (5.7)' for
V(hi)
a n d m a k i n g use of (7. o), (7.2) a n d (7.3), one finds t h a t(7.4)
J~. (f, a)arg[e,A'(a,,a)]
a ~ [A'Lao, a) 1= t ] -
[B
(ao,-) J + L (i -.:-, . . . . , n)where Ii is an integer, a n d ei---~ I when a i is a zero, a n d - - I when a,. is a pole.
Deformation Classes of Meromorphic Functions. 75
76 Marston Morse and Mauriee H e i n s . it is seen t h a t t h e a-circuit ~ induces a t r a n s f o r m a t i o n
T x ( J ) = (J) + (r)
of {J, fl} onto itself, in f a c t a translation. W e t e r m ( r ) t h e J-displacement vector induced by 4. The J-displacement vectors induced by a-circuits from (/1) to (•) form an additive abelian group ~2. The group ~2 is a subgroup of the additive group G of all i n t e g r a l vectors (r); ~ m a y coincide w i t h G, be a proper subgroup of G a n d even reduce to the null element.
We shall seek a set of g e n e r a t o r s of ~.
To t h a t end let zp and zq be a n y two distinct points of (fl). An a-circuit G(%, Zq) leading f r o m (fl) to (fl) will be defined in which M1 points of (fl) except the
t _ _ Z t
t (o < t < I) are such t h a t Zp q pair (z~,, zq) r e m a i n fixed while the paths z t, Zq = =
r o t a t e s t h r o u g h an angle - - 2 z . W e suppose, moreover, t h a t these paths lie on a topological disc on S which does n o t intersect the set (t?)--(z~,zq). Such p a t h s clearly exist. The disc can t h e n be isotopically d e f o r m e d on itself, into a disc arbitrarily close to a point. I f this a-circuit is used in (7.6), t h e only t e r m s which will m a k e a non-null c o n t r i b u t i o n are those which involve z t - - z ~ or z ~ - 4 "
The ease tt > o. I n this case we introduce the a-circuit
Z~ = a(a~,b,) (k = ~ , . . . , n) a n d obtain the following lemma.
L e m m a 7 . 1 . When ~ > o, there exist a-circuits s k = I , . . . , n, for which the components of the corresponding J-displacement vectors are d~, 1 i = I . . . . , n. These vectors ge~erate ~ as the complete group G of integral vectors (r).
The case [~ ~ o and m > 2. I n this case the a-circuits
Zrs = G (at, a,) (r < s)
are introduced. The following l e m m a results.
L e m m a 7.2. When / ~ = o and m > 2 there exist a-circuits ; ~ ( r , s = o , I , . . . , n;
r < s), for which the corresponding J-displacement vector D~.s has the components
( 7 . 7 ) - - ( i = . . . , n )
when r s ~ o, and when r = o has all components ~r except the s-th, which is zero.
The vectors D,~ generate the group ~. W h e n m is odd, ~2 is the group G o f all integral vectors (r). When m -~ 4, 6, 8, . .., s is the subgroup of G o f vectors (r) f o r which Xr~ is even.
i The Kronecker delta gives the i-th component.
Deformation Classes of Meromorphic Functions. 77 T h e c o m p o n e n t s of Dr~ are obvious f r o m (7.6).
W h e n m is odd, t h e m a t r i x whose columns are t h e c o m p o n e n t s of t h e v e c t o r s
(7.8) D1,., D2s,. . . D n - t n Do1 (m --~- n + I)
has t h e d e t e r m i n a n t w = - - I . F o r example, when m = 5:
- - I 0
- - - I - - I 0 0 - - I - - I
0 0 - - I
(7.9) r =
O0111 = O0 I
- - I - - I 00 ~ I - - I
0 0 - - I
= - - I ,
W h e n m is odd, t h e vectors a c c o r d i n g l y g e n e r a t e G a n d h e n c e $2.
T o t r e a t t h e case m = 4, 6, 8 , . . . , let us t e r m a v e c t o r (r) f o r w h i c h ~r~ is even, of even c a t e g o r y , o t h e r w i s e of odd c a t e g o r y . W h e n m is even, each vector Dr8 is of even c a t e g o r y , a n d h e n c e t h e vectors g e n e r a t e d by t h e v e c t o r s D~s are of even category.
T h e v e c t o r s D~s g e n e r a t e t h e g r o u p $2.
T o see this let )~ be an a r b i t r a r y admissible a-circuit. T h e v e c t o r a t~_
ast
(r < s) r o t a t e s t h r o u g h 2 z an i n t e g r a l n u m b e r m ~ of times (possibly zero) as t increases f r o m o to I. I t follows f r o m (7.6) t h a t the J - d i s p l a c e m e n t v e c t o r i n d u c e d by ~ has t h e f o r m
- - ~ m r s D r s
w h e r e t h e s u m m a t i o n e x t e n d s over t h e pairs (r,s) w i t h r < s. T h u s t h e vectors Drs g e n e r a t e $2.
I t r e m a i n s to show t h a t e v e r y v e c t o r (r) of even c a t e g o r y is in $2. To t h a t end we i n t r o d u c e a v e c t o r E whose c o m p o n e n t s are ~ . T h e m a t r i x w h o s e columns are t h e c o m p o n e n t s of the vectors
(7" IO) /)1~, -D~s,..., D n - l n , E
is of odd o r d e r n a n d has a d e t e r m i n a n t I so t h a t the vectors (7. IO) g e n e r a t e G.
T h u s (r) is of t h e f o r m
(7. I I) (r) : s E + D
w h e r e s is a n i n t e g e r a n d D is in ~. O b s e r v e t h a t
D,~ - - D~a + D~4 - - q- . . . . D n - l n -F D i n = - - 2 E
so t h a t s in (7. I I ) can be t a k e n as I or o. T h e v e c t o r D is of even category,
78 Marston Morse and Maurice Heins.
and if (r) is of even category, s in (7. I I) cannot be I. T h u s (r) is in ~, if of even category. This completes the proof when m ~ 2, 4, 6, 8 . . .
The case ~ - ~ o , m : 2 .
In this case there is b u t one value of i in (7.6), and J J l : O . H e n c e ~ reduces to the vector ( r ) ~ o.w 8. The generation of interior transformatio~2s by composition f v with restricted homeomorphisms 7.
W e have seen in w 6 t h a t there is at least one interior trans- f o r m a t i o n f with a prescribed characteristic set (fl)- W e shall see to w h a t e x t e n t one can choose restricted h o m e o m o r p h i s m s V of S onto itself so t h a t f 7 has invariants (J) arbitrarily prescribed from those associated with (fl).To t h a t end we first connect restricted h o m e 0 m o r p h i s m s 7 leaving (~) fixed with a-circuits f r o m (fl) to (fl).
A n y sense-preserving h o m e o m o r p h i s m 7 of S may be g e n e r a t e d as the ter- minal h o m e o m o r p h i s m of an isotopic d e f o r m a t i o n ~t of S f r o m the identity.
More explicitly there exists a I-parameter family 7] t of h o m e o m o r p h i s m s S onto S of the f o r m
(s. 7 ' = t) (o _-< =<
where 9~ is c o n t i n u o u s in z and t,
-= e o) and
L e t (a s) be the a n t e c e d e n t of (8) u n d e r 7 t. I f 7 is a restricted h o m e o m o r p h i s m leaving (~} fixed, (a t) determines an a-circuit J~ f r o m (~) to (fl). W e shall say t h a t
V induces
this a-circuit. I f f is an interior t r a n s f o r m a t i o n with the characteristic set (t?}, the composite function of z, f 7 t, affords a terminally restricted deforma- tion f~ of f in which the characteristic set of f t at the time t is (at).F o r m u l a (7.6) is applicable to the f - d e f o r m a t i o n
f t - - - f T t
with its associated~-circuit (ctt), o ~ t ~ I, and yields the result
(8.2)
Js(f7, f l ) - - J i ( f , fl)=ri
( i : 1 , 2 , . . . , n)where (r) is the J - d i s p l a c e m e n t vector d e t e r m i n e d by (at). This displacement vector is i n d e p e n d e n t of the choice of a-circuits A induced by 7 since for the same f, (if), and ~ in (8.2), a second choice of a 2 induced by 7 c a n n o t change (r).
The vector (r) is a J - d i s p l a c e m e n t vector D~(~7)
determined
by ~7 in the group ~2.I f D.2(Z) is the vector in ~
determined
by the a-circuit ~, then D~ (7) = D~ (;~)whenever ). is induced by 7.
Deformation Classes of Meromorphic Functions. 79 I t can be shown t h a t a n y a-circuit (at), o ~ t ~ I, from (fl) to (fl) is induced by some restricted h o m e o m o r p h i s m of S leaving (~) fixed; to establish this one m u s t show t h a t there exists an isotopic d e f o r m a t i o n vt of S from the i d e n t i t y in which (a t) is the a n t e c e d e n t of (~) at the time t a n d in which ~ is a r e s t r i c t e d
h o m e o m o r p h i s m leaving (fl) fixed. The details of a proof of this need n o t be given. I t is sufficient to suggest to the r e a d e r t h a t V~ can be defined by a sequence of d e f o r m a t i o n s in each of which j u s t one point of (a t ) is moved f r o m a n initial point z o to a n e a r b y p o i n t z~. One can m a k e use of a d e f o r m a t i o n J from t h e i d e n t i t y of a small circular n e i g h b o r h o o d _AT of Zo, defining the deforma- t i o n J as t h e i d e n t i t y outside of N.
We summarize as follows:
L e m m a 8 . 1 . Each restricted homeomorphism B of S leaving ([~) fixed induces a class of a-circuits ~ leading from (fl) to (fl), and every a-circuit leading from (fl) to (fl) is induced by a class of restricted homeornorphi8ms ~ leaving (fl) fixed. I f B induces ~ and (r) is the J-displacement vector determined in (7.6) by ~, then (8.2) holds for every interior transformation with (fl) as a characteristic set. The vector (r) depends only on ,~ and not on the choice of an a-circuit ~ induced by 7.
The reciprocal relations between r e s t r i c t e d h o m e o m o r p h i s m s B a n d t h e i r induced a-circuits a n d the theorems on t h e n a t u r e of t h e group ~Q of J - d i s p l a c e m e n t vectors
i n d u c e d by a-circuits yield t h e following t h e o r e m :
T h e o r e m 8. 1. I f f o is an interior transformatio~ of S with the charactel"istic set (fl) and invariants (jo), suitably chosen restricted homeomorphisms B of S leaving (fl) fixed will yield interior tran.~formations f o B with invariants (jo) + (r) where
(I) (t-) is an arbitrary integral vector when t~ > o , or when t*-~o and m is odd, (2) (1") is an arbitrary integral vector of even category when I~ = o and m - ~ 4 , 6 , 8 , . . . ,
(3) (r)----(o) only, when t*-~ o and m = 2.
No other values of (J) can be obtained by composition f o ~ of fO with restricted homeomorphisms ~.
w 9. a-circuits rood (fl) and semi-restricted homeomorphisms ~ B of S. W e resume the t h e o r y of admissible a-circuits mod (fi) i n i t i a t e d in w 7. As in w 7 we a r e
concerned w i t h an admissible d e f o r m a t i o n )it, o ~ t_--_ < I, f o r which (a t) is the characteristic set of f t a t the time t. W e suppose t h a t (a ~) is a n admissible re- o r d e r i n g of (a~ I n p a r t i c u l a r a~o---a ~
The present section could be omitted by a reader who wishes to comprehend first the main theory.
80 Marston Morse and Mauriee Heins.
Such a r e o r d e r i n g defines a p e r m u t a t i o n ~r of ( x , . . . , n) in which i is replaced by ~(i) and
, o (i I, n).
(9"
I)
a~ = a~ (i) ~- . - . ,F o r any interior t r a n s f o r m a t i o n f with characteristic set (a~
( 9 . 2 )
J,(A
, ' ) = J , ( , ) ( f , ~o) (r = ~,..., ,)in accordance with the definition o f (J). I f ( x x , . . . , x,) is an a r b i t r a r y set of n symbols, we shall write
(x~(1) . . . . , x , ( , ) ) = ~ ( x ) . T h u s (9.2) takes the f o r m
[J(f,
a~)] = z [ J ( f ,a~
I t follows f r o m (7.6) t h a t
(9.3)
j,(f,,
a a) ___ j . ( f 0 , a0) + O,(~) [Z -~ (at)].From (9.2) and (9.3) one sees t h a t
(9.4)
j,(,) (f~, fl) : j , (fo, fl) + 0,(~)
(fl) -~ (a~E q u a t i o n s (9-4) may be written in the vector forms (9.5) ~r [ J ( f ' , fl)] = [ j ( f 0 , fl)] + [0].
(9.6) [ j (f~, fl)] ___ z~-i { j ( f 0 , fl) + [0]}.
W e thus have the following lemma:
Lemmn. 9 . 1 ,