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The topology of reduced phase spaces of the motion of vortices on a sphere
Physica D 30 (1988) 99-123 North-Holland, Amsterdam
T H E T O P O L O G Y OF REDUCED P H A S E SPACES OF T H E M O T I O N OF VORTICES ON A S P H E R E Frances K I R W A N Mathematical Institute, 24-29 St. Giles, Oxford, UK
Received 10 February 1986 Revised manuscript received16 November 1987
The motion of point vortices in a perfect incompressible fluid on a two-dimensional sphere can be represented by a Hamiltonian flowwhich is invariant under the action of the special orthogonalgroup SO(3). Followingthe reduction method of Maxsden and Weinstein this motion can be described by Hamiltonian flows on reduced phase spaces. Some topological invariants of these reduced phase spaces are calculated. For suitable vorticitiesthe results obtained can be used to give lower bounds on numbers of relativeequilibria of the original system. Introduction In [6] Bogomolov models the motion of n cyclones on the earth as a Hamiltonian flow on a phase space which is the product of n copies of a two-dimensional sphere S 2 with a symplectic form depending on the vorticities of the cyclones. (This simple model depends on the assumption that large scale motions in geophysical fluid dynamics are approximately two-dimensional and disregards the earth's rotation.) Both the symplectic structure and the Hamiltonian are invariant under an action of the special orthogonal group SO(3) and hence there is a momentum map /~ whose values are invariants of the flow. Thus following Marsden and Weinstein [12] the motion can also be described for generic values of the vorticities by the associated Hamiitonian flow on a reduced phase space. The aim of this paper is to study the topology of these reduced phase spaces by computing some topological invariants, their Betti numbers (see (3.9) and (3.11)). Knowledge of these gives one some control over, for example, the number of equihbrium points of the reduced Hamiltonian flow when the reduced Hamiltonian is a Morse function. This is also the number of orbits in the original phase space consisting of stationary or uniformly rotating configurations of vortices with a given value ~ ~ R 3 of the invariant function #. We shall obtain a lower bound for this number, which depends on ~ and the vorticities (see theorem 3.16). Of course for some vectors ~ in R 3 there are no configurations at all, uniformly rotating or otherwise, at which # takes the value ~, and in these cases the lower bound is necessarily zero. On the other hand, we shall see that for suitable vorticities and suitable ~ the lower bound becomes large rapidly as the number of vorticities becomes large. The layout of the paper is as follows. Section 1 describes Bogomolov's model and the reduced phase spaces in more detail. Section 2 briefly recalls the definition of the Betti numbers of a manifold and why they are related to the numbers of fixed points of Hamiltonian flows. Section 3 gives the main results and an outline of their proof. Section 4 consists of general nonsense: it gives a general procedure for computing the Betti numbers of reduced phase spaces which is used to prove the assertions made in section 3. I would like to thank J. Koiller for suggesting this problem to me (see also [11] section 12) and for helpful advice. He also pointed out to me that a similar approach to a different problem is taken by PaceUa in [14]. 0167-2789/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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1. Point vortices on a sphere
The motion of n point vortices on a two-dimensional sphere flow on the phase space x=s
S 2 can
be represented by a Hamiltonian
2 x - - - x S 2 = ( s 2 ) ".
For simplicity we assume that S 2 is the unit sphere in R 3 The symplectic form on X is
to=
E
kjto ,
l <_j<_n
where % is the pullback to X via the j t h projection onto S 2 of the standard symplectic form on S 2, and kj is the vorticity of the j t h vortex. The Hamiltonian function H is defined as follows. If xj ~ S 2 _ ~3 for 1 < j < n then
H(xi,...,x,,)=-
E
kikjlog(1-x~'xj),
l <_i
where xi • x j is the standard inner product of the unit vectors x i and xj in R 3. (See [5]. For a survey article on vortices see [2].) 1.1. Remark. T h e Hamiltonian function H blows up along the diagonals of X (i.e. when x i = xj for some i ~ j ) . For the time being we shall ignore this problem in order to work as long as possible with compact spaces without boundary. Eventually we shall remove the diagonals (see section 3). The precise form of the Hamiltonian will not in fact matter greatly to us. Its only important property is that it is invariant under the action of the special orthogonal group SO(3) on X. Let the special orthogonal group SO(3) be denoted by K. This group acts naturally on S 2 (as a group of rotations) and hence also on X. It is easy to check that both the symplectic form to and the Hamiltonian function H are invariant under the action of K on X. Let / be the Lie algebra of K and let xe* be its dual space. (Recall that xe consists of all skew-symmetric 3 × 3 real matrices.) The Killing form defines an inner product on xe which is invariant under the action of K on ,( by conjugation. We can identify both / and 1" with R 3 using an orthonormal basis; under these identifications the action of K on / and / * given by conjugation becomes the usual action of K on R 3. Because the symplectic structure on X is preserved by K there is a momentum mapping g: X - , / *
= R3
determined by the following two properties: 1.2. (i) # ( k x ) = k g ( x ) for all x ~ X and k ~ K. (Here k x is the result of acting on x ~ X by k ~ K and k g ( x ) is the result of acting on g ( x ) ~ / * = R 3 by k ~ K). (ii) Given any a ~ / there is a function ga: X--, R which sends x ~ X to the natural pairing g ( x ) • a of g ( x ) ~ g* with a ~ xe. There is also a vector field x ~ ax on X defined by the infinitesimal action of K on
F. Kirwan / Motion of vortices on a sphere
101
X, i.e. d ax = -dT ( exp ( ta ) x ) t_ o. This vector field is the Hamiitonian vector field of the function/£a" That is, d # a ( x ; ~/) = tox(71, a~)
(1.3)
for all x ~ X, ~ ~ TxS and a ~ ~¢. (See [1] 4.2 or [9] section 2 or [12] for more details on momentum mappings.) In our case the momentum mapping St/,: X---- ( s 2 ) n " ~ g f* = R 3
is given by
,(xl,...,x.)= E kj j, l
when ~¢* is identified with R 3 and S 2 is identified with the unit sphere in R 3 (see [2] p. 322). The Hamiltonian function H is invariant under the action of K = SO(3) so d H ( x ; ax) = 0 for all x ~ X and a ~ A¢. By (1.3) this is equivalent to saying that the momentum mapping g is constant along the flow of the Hamiltonian vector field of H, i.e. the flow preserves #-l(/j) for all/j ~ k* = R 3. If ~ ~ xe* = R 3 then #-1(~) is invariant under the subgroup S t a b / j = { k ~ KIk/j = /j } of K. It is easy to check that when/j is nonzero Stab/j is isomorphic to the circle group S x (when xe* is identified with R 3 then Stab/j is the group of rotations about the axis in R 3 defined by/j) whereas when /j = 0 then Stab/j is the whole group K = SO(3). The reduced phase space, or Marsden-Weinstein reduction, associated to any element /j of xe* is the quotient space Xt~= / x - l(/j)/Stab/j = t t - I ( K / J ) / K , where K/J is the orbit of/j (see [12]). If Stab/j acts freely on/~-l(/j) then/j is a regular value of # and X~ is a compact manifold of dimension dim X~ = dim X - dim K - dim Stab/j. The property (1.3) of a momentum map implies that when restricted to #-l(/j) the symplectic form to becomes degenerate precisely along the directions tangential to Stab/j orbits. Thus to induces a symplectic form to~ on the reduced phase space X¢ (provided that/j is a regular value of/~). Since the function H is K-invariant it induces a function He on X¢ and the Hamiltonian flow of H~ lifts to the Hamiltonian flow
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of H restricted to F-I(~). In particular the equilibrium points for the Hamiltonian flow of H~ on X~ correspond to those solution curves in #-1(~) of the original system which are orbits of one-parameter subgroups of K (see [3] appendix 5), i.e. to configurations of vortices which either rotate uniformly about some axis in R 3 or are stationary. The aim of this paper is to compute the Betti numbers of the reduced phase space X~ when ~ is a regular value of # and to use these to obtain information on the number of equilibrium points for the Hamiltonian flow of H~ on X~. (Since H blows up along the diagonals in X, we shall actually be interested in equilibrium points not contained in the diagonals.) The Betti numbers of a manifold are topological invariants; in the next section we recall their definition and how they are related to the number of equilibrium points of a Hamiltonian flow. For other examples of the application of Marsden-Weinstein reduction to problems involving vortices, see e.g. [10, 11].
2. Betti numbers and Morse theory We recall the definition of the Betti numbers
b o ( Y ) , b l ( r ) , b2(Y) . . . . of a topological space Y. These are nonnegative integers which contain information about the topology of Y. A singular i-simplex in Y is a continuous map a: A t ~ Y where A i is the convex hull of the i + 1 points (0 . . . . . 0 ) , ( 1 , 0 . . . . . 0),(0,1,0 . . . . . 0) . . . . . (0 . . . . . 0,1) in R i. The vector space Ci(Y; R) of singular /-chains with real coefficients consists of all formal linear combinations
ala t + . . . + apav of singular i-simplices with real coefficients aj ~ R. A singular ( i - 1)-simplex is a face of a singular /-simplex a if it is the composition of a: A i ~ Y with one of i + 1 standard maps identifying Ai_ 1 with the faces of A r The boundary of a is defined by
aa=
~
+~,
1" face o f o
where the signs depend on a suitable choice of orientations. We can extend O by linearity to a map a: ci(Y; R) --, C,_l(Y; R ). Then a 2= 0. The (singular) homology groups of Y with real coefficients are now defined to be the quotient vector spaces Hi(y)=
( *1e C / ( Y ; R ) I 0 n = 0}
F. Kirwan / Motion of vortices on a sphere
103
If Y is a reasonably well-behaved space (e.g. a manifold) then these vector spaces Hi(Y ) are finitedimensional and the Betti numbers of Y are defined by
bi(Y ) = dim Hi(Y ). For example when Y-- S 2 then 1, 0,
bi(Y) =
i = 0,2, otherwise.
(2.1)
The cohomology groups Hi(y) of Y with real coefficients are the duals of the homology groups Hi(Y ). When Y is a compact manifold they are the same as the De Rham cohomology groups of Y. The information given by the Betti numbers is encoded in the Poincar6 series
Pt(Y) = y' bi(Y)t i i>O
of Y. If Y is a manifold of dimension m then bi(Y ) = 0 if i > m and so we can define the Euler number
x ( Y ) of Y by
(2.2)
x(Y) = ~, ( - 1 ) i b i ( Y ) = P _ l ( Y ) . i>O
It has the nice property that if Z is a submanifold of codimension y in Y then
(2.3) This can be deduced easily from the existence of a long exact sequence
... ~ H i - ~ ( Z ) ~ H i ( Y ) ~ H i ( y - Z ) ~ H i - r + l ( z ) ~
...
(2.4)
called the T h o m - G y s i n sequence. (See e.g. [15] for more details.) Now suppose that Y is a compact manifold and f: Y ~ R is a smooth function. By perturbing f an arbitrarily small amount we may assume that f is a Morse function, i.e. every critical point x ~ Y for f is isolated and nondegenerate in the sense that the Hessian of f at x (which is given in local coordinates by the matrix of second partial derivatives of f at x) has maximal rank. The index i/(x) of f at x is then defined to be the number of strictly negative eigenvalues of the Hessian of f at x. The Betti numbers of Y are related to the number of critical points of f (i.e. the points at which the derivative of f vanishes) by the famous Morse inequalities, which can be expressed most easily in the form
~., t i / ( x ) - P t ( Y ) = R ( t ) ( l + t ) ,
R(t)>O.
(2.5)
x critical
That is, R(t) is some polynomial in t with nonnegatioe integer coefficients. In special cases (when f is a perfect Morse function) R(t) = 0 so the Betti numbers bi(Y ) are equal to the numbers of critical points of f with given index i. In general putting t = - 1 we get
x(Y)=P_x(Y) =
~, x critical
(-1)/:(x),
(2.6)
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F. Kirwan / Motion of vortices on a sphere
so the total number of critical points is always at least the modulus
Ix(Y)l
(2.7)
of the Euler number of Y. The Morse inequalities can be derived by putting a Riemannian metric on Y and looking at the induced gradient flow of the function f on Y. The path of steepest descent of any point of Y converges to a critical point for f , and we can stratify Y by putting points into the same stratum if and only if their paths of steepest descent converge to the same critical point. The codimension of the stratum corresponding to a critical point x is the index if(x). By applying the T h o m - G y s i n sequence to each stratum in turn we obtain the Morse inequalities. For alternative arguments see [13] and [16]. If the critical points of f are not all isolated then the total number of critical points is infinite and so trivially greater than Ix(Y)I. However, if the critical points form submanifolds of Y and the Hessian is nondegenerate in directions normal to these submanifolds, then we can obtain more sophisticated Morse inequalities
~,ti:(C)pt(c ) - Pt(Y) = R ( t ) ( 1 + t),
R ( t ) > O,
where the sum runs over all the connected components C of the set of critical points and if(C) is the index of f at any point of C. If Y is not compact, then the Morse inequalities may fail (for example, simply remove all the critical points from Y). However, if Y is a compact Riemannian manifold and Z is a closed subset and f: Y - Z ~ R is a Morse function such that the path of steepest descent of every point of Y - Z converges to a point of Y - Z, not Z, then the proof of the Morse inequalities still holds and hence the total number of critical points of f is at least
Ix(r- z)I.
(2.8)
In particular, these conditions are satisfied if f ( y ) ~ oo whenever y ~ Y - Z approaches Z. The Morse inequalities together with (2.3) can also be used to show that if f: Y - Z ~ R attains its minimum value on a closed submanifold W of Y - Z then the total number of critical points of f in Y - Z - W is at least Ix(Y-
z) - x(W)[ =
Ix(Y- z - w)I.
(2.9)
We can apply this to study the number of critical points of the Hamiltonlan function H~ defined in section 1 (or equivalently the number of equilibrium points of the Hamiltonian flow). Let A = ( ( x x. . . . . xn) ~ X = (S2)nlXl = xj for some i :~j ) and
AS = (A N #-l(~))/Stab ~ c X~. Let X ° -- X - A and X~° = X~ - A~. Then H: X ° -) R is defined by
H ( x 1. . . . . x . ) = -
• l
kikjlog(1-x,.xj)
F. Kirwan / Motion of vortices on a sphere
105
and induces H,: X~° ~ R. Let A += ( ( x t ..... xn) ~ XIx i = xj, some i * j such that kik j > 0} and
a [ = (a+n # - t ( ~ ) ) / S t a b ~ ___X~, Then the function exp H: X - A + ~ R defined by exp H ( x 1. . . . . x , ) =
1-I
(1-xi'xj)
\ -kikj
l
is a smooth function invariant under the action of K--SO(3). Hence it induces a function expH~: X ~ - A T ~ R which takes its minimum value 0 precisely on A ~ - za~, and its other critical points are exactly the critical points of H~: X~° --, R. Moreover it is easy to check that a path of steepest descent for exp H from any point of X~ - A~- cannot converge to za~- and hence its limit points lie in X~ - A~-. Since
the argument for (2.9) shows that provided the values k i of the vorticities are such that the Hamiltonian function H~ is a Morse function on X,°, the number of its equilibrium points is at least
Ix(x,0)l
(2.10)
In the next section we shall see how this Euler number x ( X ~ ) can be calculated using equivariant Morse theory. So to complete this section we need a few remarks about equivariant cohomology and equivariant Morse theory. For this we assume that a compact group K acts on Y. Let
E K --* B K be the universal classifying bundle of K. Then EK is a contractible space on which K acts freely and BK is the quotient of E K by K. We define the equivariant cohomology groups H~c(Y ) of Y to be the ordinary cohomology groups of the quotient
Y ×r E K of Y × E K by the diagonal action of K. The point of this is that if K acts freely on Y (or more generally if every x ~ Y is fixed by only finitely many elements of K ) then
Hk(Y) = Hi(y/K),
(2.11)
but in general H~c(Y) has better properties than H i ( y / K ) . If f: Y ~ R is a K-invariant Morse function we can do equivariant Morse theory replacing ordinary cohomology by equivariant cohomology throughout, and we obtain equivariant Morse inequalities relating
F. Kirwan ~Motion of vortices on a sphere
106
the equivariant Poincar6 series
Ptk(Y) = E tidimHk(Y) i>O
of Y to the equivariant Poincar6 series and indices of the connected components of the set of critical points for f.
3. The results
We saw in section 1 that the momentum mapping #: X ~ / * X = ($2) n with symplectic form
Ekj
l
= R 3 for the action of K = SO(3) on
j
is given by
~(x 1..... x,)=
E ~xj. l
We are interested in computing the Betti numbers of the reduced phase space X¢ = # - l ( ~ ) / S t a b when ~ is a regular value of #, in order to obtain bounds for the number of fixed points of Hamiltonian flows on X~ (see section 2). The idea is to apply equivariant Morse theory to the function f : X--,R defined by f(xl .....
x=) - 6 II=.
Here II II denotes the usual length function on R 3 which is invariant under the action of K = SO(3), so f is invariant under the action of Stab ~. It turns out (see (4.14)) that the function f is equivariantly perfect in the sense that its equivariant Morse inequalities are actually equalities. In other words the equivariant Poincar6 series of X with respect to the action of Stab ~ satisfies PtStab~( x ) = Eti'(c)PtStab~( c ) ,
(3.1)
where the sum is over all connected components C of the set of critical points for f, and if(C) denotes the index of f along C. (That is, i/(C) is the number of strictly negative eigenvalues of the Hessian of f on the tangent space to X at any x ~ C.)
F. Kirwan/ Motion of vortices on a sphere
107
It is clear that f attains its minimum value precisely on the subset #-1(4) of X. Therefore we can rearrange (3.1) to get
PtSUa'~(#-a(4))=PtStab~(x) -
E
tq(C)p,S'~'~(C) •
(3.2)
c¢~,-1(~) By assumption 4 is a regular value of #, i.e. if x ~ #-1(4) then the derivative of # at x has maximal rank. Because of the defining property (1.3) of a momentum map this means that any x ~ / ~ - l ( x ) is fixed by only finitely many elements of Stab 4. Therefore by (2.11) the Poincar~ polynomial of X~ = # - 1 ( 4 ) / S t a b 4
satisfies Pt(X~) = P t S ~ ( # - l ( 4 ) )
(3.3)
and hence is given by the right-hand side of (3.2). On the other hand, because X is a compact symplectic manifold and the group action preserves its symplectic structure we have PtStab£(X) = Pt( X)Pt(B(Stab 4)),
(3.4)
where B(Stab 4) is the classifying space of Stab 4 (see [9] 5.8). If 4 is nonzero then Stab ~ is isomorphic to the circle group S t and
Pt(B(Stab 4)) = Pt( B $1) = 1 + t 2 + t ' + . . . .
(1 - t 2)-1.
(See e.g. [5].) If 4 = 0 then Stab 4 = S0(3) and
Pt(B(Stab 4)) = Pt(B S0(3)) = 1 + t ' + t 8 + . . . .
(1 - t ' ) - l .
Moreover
Pt(X)=(Pt(S2))"=(I
+t2)"
Thus ptSt~b~(x) =
( 1 + t 2 ) " ( 1 - - t 2 ) -1, ( 1 + t 2 ) " - l ( 1 - - t 2 ) -1,
if4*0, if4=0.
(3.5)
Finally we must deal with the terms
tiI(C)PtStab~( c ) in (3.2). Note first that if 4 and [ lie in the same K-orbit in ~¢* = R 3 then the reduced phase spaces X~ and
F. Kirwan / Motion of vortices on a sphere
108
X~ are the same: X~ = # - l ( ~ ) / S t a b ~ = ~ - I ( K ~ ) / K = # - I ( K ~ ) / K = X~. Hence we may assume without loss of generality that ~ is of the form = (0,0, I1~11).
(3.6)
Then it turns out (see (4.4)) that when ~ is nonzero the critical points of f~ not contained in #-1(~) are the points .... ,
x=
satisfying xl -- (0, 0, + 1) for 1 < i _< n. There are exactly 2 n such points, one for each subset S of {1 ..... n }. We write
xS = ( x s .... , x s) where [ (0,0,1), xS= ~ (0,0,-1)
if i ~ S ,
ifi~S.
If ~ = (0,0, [1~11)is nonzero then Stab ~ is the group of rotations about the third coordinate axis in R 3, and hence Stab ~ fixes each point x s of the form above. The equivariant Poincar6 series of any single orbit is always Pt(BY~) where ~ is the stabiliser of a point in the orbit (see e.g. [5] section 13). In this case = Stab ~ and we get
ptStab*( { x S ) ) = Pt( B(Stab ~ ) ) = Pt( BS 1) = (1 - t2) -1. Thus combining (3.2), (3.3) and (3.5), when ~ is nonzero we get
Pt(X~) = (1 + / 2 ) n ( 1 - t2) - 1 -
E
tx(s)(1 - rE) -1,
(3.7)
SC_{l . . . . . . )
where X(S) is the index of f at the point x s. It is shown in section 4 that this index can be computed as follows. If S is a subset of {1,..., n } let k(S):
Ekjj~S
j~S
and
c(S)= #{jESlkj>O
} + #{jq~S[kj
F. Kirwan/ Motionof vorticeson a sphere
109
where # F denotes the number of elements of any finite set F. Then if ~ q: 0 is a regular value for # and S is a subset of {1 . . . . . n} we have k(S) ~ I1~11and
~(s)
[ 2c(S), [ 2(n -
c(S)),
if
k(S)
if
k(S) <
> II~ll,
(3.8)
II~ll-
(See (4.17).) Combining (3.7) and (3.8) we obtain 3.9. Theorem. Let ~ be a nonzero regular value of/~. Then the Poincard polynomial phase space X~ is
(1--t2)-1[ (l + t2)n- k(S)Y'~II~ll Examples. (a) If the vorticities + • • • + k . - 2 rain kj we have
3.10.
Pt(X~) of
the reduced
k(s)Y]~I~l t2(n-c(S))] k I . . . . , k , are all positive, then when k I + . . . + k , > I1~11> kl
k(S) > I1~11"~S = ( 1 , . . . , n }, and hence
Pt(Xt~)=(a-t2)-2[(l+t2)n-t2 k 2 > have
2 + t a + "-" + t 2("-2). "" >k,>O
then when k l + . . .
+k,_l-k,>ll~ll>kl + ... + k , _ 2 - k , _ x + k . we
k(S)>ll~ll**S={1,...,n} or {1 .... , n - l } , and then
Pt(X~) = (1
- t2)-111 + t 2 -- t 2 ( n - l ) - - / 2 ( n - 2 ) ]
= 1 + 2 t 2 + 2t 4 + • • • + 2 t 2(n-3) + t 2(n-2). N o t e that as in (a) this is a polynomial of degree d = 2n - 4 in t with nonnegative coefficients and the coefficient of tJ is the same as the coefficient of t a-j. Any compact manifold of dimension d must have a Poincard polynomial of this form, by Poineard duality. We can deal with the ease ~ = 0 similarly. The result is as follows (see section 4 for the proof). 3.11.
Theorem.
If 0 is a regular value of # then the Poincard polynomial
Pt(Xo)
of the reduced phase
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F. Kirwan / Motion of vortices on a sphere
space X 0 is
(1-- t2)-l[(l + t2)n-l-- ~" t2(c(S)-l)]. k(S)>O
J
Example. If the vorticities k x. . . . , k~ are all positive and sufficiently close together then
3.12.
k(S) > 0 "~ c(S) > n/2 and hence
Pt(Xo) = (1
-/2)-1[(1
"q- / 2 ) n - 1
_
[
n>r>n/2
= 1 +b2t2q - - ' - + b 2 j t 2 J +
...
-[-t 2(n-3),
where
b2j=l+(n_l)+(n-l)+...+(
n-1 ) min(j, n- 3-j) "
2
By theorem 3.9 if ~ is a nonzero regular value o f / t then the Euler number x(X~) of the reduced phase space X~ is x(X
) =
=
=
lim
(~'k(S)'ll~llt2(c(S)-l)q- ~k(S)
t2-_.l
( t 2 - 1)
E
(c(S)-l)+
k(S) > I1~11
E
(n-c(S))-n2
n-'
(3.13)
k(s) < I1~11
By theorem 3.11 when ~ = 0 we get
X( Xo) = P-I( Xo) = lim (Ek(S)>°t2~s)-l~-- (1 + t2) "-1) t2.... 1
= •
( t 2 - 1)
(c(S)- 1)- (n- 1)2n-2.
(3.14)
k(S)>0
We noted at (2.7) that Ix(Se)l is a lower bound for the number of critical points of a real valued Morse function on X~, and hence also for the number of fixed points of a Hamiltonian flow on X~ defined by a Morse function. Unfortunately, the Hamiltonian H e defined in section 1 blows up along the image A~ in X~ of the diagonals of X = ($2) n. We saw at (2.10) that to obtain a lower bound for the number of fixed points in X~° = X~ - A~ of the Hamiltonian flow on X~° induced by H~ we must compute
Ix(x,°tl
F. Kirwan / Motion of vortices on a sphere
111
For this we divide X~ into subsets indexed by equivalence relations on (1 . . . . , n }, or equivalently by partitions of {1 . . . . . n}. (Recall that to an equivalence relation corresponds the partition defined by its equivalence classes.) If ~r is a partition of {1 .... , n } let A~-- {(xx . . . . . x , , ) ~ X [ x i = + ' ~ i - , j } and A'~ = ( a " Cl/~-l ( ~ ) ) / S t a b ~ __ X,, where - , , is the equivalence relation corresponding to or. Note that if ¢r0 denotes the trivial partition {{1}, [2} . . . . . {n}} then
a?= xg, and
~ r ~ c ro
Moreover X~ is the disjoint union of the A~ which are locally closed submanifolds of even codimension in X~. Hence by (2.3) x(X~°)=x(X~) -
E x(A~) •
(3.15)
crier o
The reduced phase space X~ depends on n and the vortices k 1. . . . . k n as well as on £. Let us write
X~( k 1..... k,,),
X~( k 1..... k,,)
for X~ and X~° when it is necessary to demonstrate the dependence on n and k 1. . . . . k. explicitly. Then if
rr= { A~,..., Aq } is any partition of {1 . . . . . n } into disjoint nonempty subsets A~,...,
Aq it is easy to see that
A'~ _---x~o(kl, . . . . , k~(,,)), where n (~r) = q and
ky=Ek,. i~Aj
Thus (3.15) gives an inductive formula for X(X~°) in terms of X(X~) (which we have already computed: see (3.13) and (3.14)) together with terms of the same form as x(X~°) but with smaller values of n.
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F. Kirwan/ Motion of vorticeson a sphere
3.16. Theorem. If the vorticities k i are such that the reduced Hamiltonian H~ is a Morse function then the number of equilibrium points of the Hamiltonian flow of H~ on X~° is at least
where the Euler number X (X~°) = X( X~°(kl . . . . . k,)) is given inductively by the formulas
x ( g : ( k l .... ,kn)) ~- x( g~(ka ..... kn)) x(X¢(kx,...,k,))=
~.,
(c(S)-
1) +
k(S) > IIEI if ~ 0
E x ( S : ( k ' ( ..... k:(~r,)), ~rq-~ro E
(n-c(S))
-n2"-1
k (S) < I1~11
and
x ( g o ( k l ..... kn)) =
E
( c ( S ) - 1) - ( n -
1)2 n - 2 .
k(S)>0
This theorem gives us a lower bound for the number of stationary or uniformly rotating configurations of vortices on S 2 with a given value f of the momentum mapping. However, the formulas are rather complicated and it is not immediately clear that they can tell us anything interesting, so let us consider some special cases. Of course when f does not lie in the image of/~ then X~ = q~ and the lower bound we get has to be zero. This happens if and only if either (a)
II~ll>lkll+lk21+-.-+lk,I
(3.17)
or there exists some j ~ ( 1 , . . . , n ) such that (b)
II~ll
< Ikjl- E Ikil • i ~j
If equality holds in either (a) or (b) then $ is not a regular value of/~, except in the trivial case n = 1 when the lower bound is
x(Sg) =x(S~)=
( O' 1,
II~n=~:]kll'
II~ll=lkxl.
F r o m now on we shall assume that ~ is nonzero and
Ikjl- ~lkil
< l k a l + "'" +tk.I
for all j. When n > 2 the dimension of S~ is 2n - 4. When n = 2 X~ is a single point (which is automatically an equilibrium point of the flow). This is consistent with the calculations
Pt(Xf) = ( 1 - t2)-1((1 + t 2 ) 2 _ t 2 _ t 2 _ t 2 _ t4) = 1
F. Kirwan ~Motion of vortices on a sphere
113
and
x(X,)-- 1. When n = 3 we find
Pt(X~) = (1 - t2)-1((1 + t2) 3 - t 4 - 3t 2 - 3t 4 - t 6) = 1 + t 2, so
x( X~) = 2 and x(Xg)=2-x(Xg(kl,
k 2 + k 3 ) ) - x ( X g ( k 2 , kl + k 3 ) - x ( X g ( k 3 , kl + k 2 ) ) .
Thus x(X~°) = 2,1,0 or - 1 depending on whether none, one, two or all of the conditions (i)
[ I k l l - Ikz + k311 < I1~11< Ikll + Ik2 + k31,
(ii)
Ilk21- [kx + k3ll < I1~11< Ik21 + Ikx + k31,
(iii)
Ilk31-Ik~ + kzll < I1~11< Ik31+ Ikl + kzl
are satisfied. When n > 3 there are too many different cases to be listed here. However, in the special case when the vortices kl . . . . . k n are all positive and I1~11is strictly less than but close to k 1 + . . . +k n we have already seen at (3.10) that
x( X~) = P_I( X~) = n - 1 . It follows from (3.16) that
X(X~)=h(n), where
h(n) is defined inductively by
h(1)=0,
h(n)=n-1-
~_,
p(n,k)h(k)
l
and p(n, k) is the number of partitions of {1 . . . . . n } into k nonempty subsets. Thus h(2) = 1, h(3)=2-3.1=
-1,
h(4) = 3 - 7 . 1 - 6 . ( - 1 )
=2,
h(5) = 4 - 1 5 . 1 - (10 + 15). ( - 1 ) -
10.2 = - 6 ,
h(6) = 5 - 3 1 . 1 - (15 + 6 0 + 1 5 ) - ( - 1 ) -
(45 + 2 0 ) - 2 - 15-(6) = 89.
E Kirwan / Motion of vortices on a sphere
114
This shows the lower bound Ih(n)l on the number of stationary or uniformly rotating configurations increasing rapidly as n increases.
4. General n o n s e n s e
In this section we shall prove the assertions made in the calculation of the Poincard polynomials Pt(X~) in section 3. We shall work in the following more general setting. Let X be a compact symplectic manifold with symplectic form to and let K be a compact connected Lie group acting on X in such a way that the symplectic structure is preserved. Let t be the Lie algebra of K and let t * be its dual. Suppose that t~: X - , t * is a momentum map for the action of K. This means that for any a ~ 1 the component of /t in the direction of a is a Hamiltonian function for the vector field x --, a x on X induced by a; that is d # ( x ) ( ~ ) - 1= tox(~i, a x ) for all x ~ X and ~ ~ TxX; and moreover I ~ ( k x ) = Ad* k l ~ ( X )
for all k ~ K and x ~ X where Ad* is the co-adjoint action of K on l * (see e.g. [12], [1] 4.2 or [9] section 2 for details about momentum maps). Then the reduced phase space, or Marsden-Weinstein reduction, associated to any element ~ of l * is the quotient space x¢ =
where Stab ~ is the stabiliser of ~ under the co-adjoint action of K on 1 * (see [12]). If Stab ~ acts freely on g - i ( ~ ) then ~ is a regular value of # and X~ is a compact manifold of dimension dim X~ = dim X - dim K - dim Stab ~. Let us fix a K-invariant inner product on 1 (this also gives an invariant norm I1 II on l ) and use it to identify l * with t . In [9] equivariant Morse theory is applied to the function IINI2 on S to obtain a formula for the Betti numbers of the reduced phase space X 0 = / ~ - i ( 0 ) / K provided that 0 is a regular value of #. In this section we shall apply similar ideas to the function f = Ib - 6112 to obtain a formula for the Betti numbers of X~ for any ~ ~ t * = 1 which is a regular value of/x. First let us consider the critical points of the function f. For the time being we shall not need to assume that ~ is a regular value of #. Let stab ~ = ( a ~ l l [ a , 6] = 0) be the Lie algebra of Stab ~. If a ~ 1 let x ---, a x denote the vector field on X induced by a. 4.1. L e m m a . d r ( x ) = 0 if and only if # ( x ) = ~ + fl where fl ~ stab ~ and fix = 0.
F. Kirwan/ Motion of vortices on a sphere
115
Proof. The proof of [9] lemma 3.1 shows that d f ( x ) = 0 if and only if (g(x) - f)x = 0. In fact it follows easily from the definition of a momentum map that
d f ( x ) ( * / ) = ~x(7/,2(la(x) - f ) x ) for every x ~ X and 71~ T~X. If .8 = #(x) - f and .8~ = 0 then [/3, g(x)] = 0 because/~ is a K-equivariant map. Hence
0 = [ . 8 , . 8 + f ] = [.8,f], s o ,8 ~ stab f.
[]
Suppose that .8 is an clement of xe. Let Ta be the subtorus of K generated by .8. Then the inner product g ( x ) . . 8 of g ( x ) and .8 is constant on every connected component of the fixed point set FIX(Ta) of Ta in X (see e.g. [9] 3.7). Let Za be the union of those connected components of Fix(T~) on which g ( x ) . / 3 = (.8 + f ) " .8. Then Z~ is a Stab.8-invariant symplectic submanifold of X ([8] 3.6 or [9] 4.8). Let T be a maximal torus of K with lie algebra t such that f ~ ~'. Then the orthogonal projection /~r: X -o , ~ [ * of tt onto d is a momentum mapping for the action of T on X ([9] 3.3). The image under tt r of the fixed point set of T is a finite subset ~¢ of ~'. It has been proved by Atiyah in [4] and by Guillemin and Sternberg in [8] that g T ( X ) = Conv~¢, the convex hull of a¢ in ~. For each nonempty subset ~¢' of a¢, the convex hull Conv ( a - fla ~J¢'
} = Conv
'- f
contains a unique point .8(a¢') which is closest to the origin of ¢' (i.e. at which the norm takes its minimum value). Let ~ r be the finite subset of ¢' consisting of all such points .8(z¢'). Let g = ~ r n ~'+ where ~'+ is a fixed positive Weyl chamber for Stab f in ~. (Note that T is a maximal torus of Stab f as w e l l as K.)
4.2. Lemma. Suppose that d f ( x ) = 0. Then there exists k ~ Stab f and .8 ~ ~ such that g ( k x ) - f = .8. Proof. By (4.1) g ( x ) - f = .8 where .8~ = 0 and .8 ~ stab1. Therefore there exists k ~ Stab1 such that Ad(k).8 ~ ~+. Since g ( k x ) - f = Ad ( k ) g ( x ) - f = Ad (k).8,
we may thus assume that .8 ~ ~+ and it suffices to show that fl ~ ~ . We have x E Z# because .8~ = 0 and tt(x)..8 = (.8 + f ) " .8. Moreover g ( x ) = gT(X) because g ( x ) ~ ~'. Therefore by the theorem of Atiyah and Guillemin-Sternberg ([9] 3.4) applied to the action of T on Za we have .8 = ~ ( x )
- f ~ ~T(z~)
- f = Conv~'
- f,
where ~¢' is the subset gr(Z[~ t3 Fix T ) of ~¢=/.tT(Fix T). Furthermore p-(~T(x)
- f ) = 11.sll 2
F. Kirwan/ Motion of vortices on a sphere
116
for all x ~ Zp, so if 71~ Conv.a¢' - ~ then so fl ~.~'. [] If f l ~
117/11>--IIBII with equality if and
only if n = ft. Hence fl = fl(~¢')
let Ca= S t a b ~ ( Z o n / ~ - l ( f l + ~)).
4.3. Corollary. The set of critical points for f on X is the disjoint union of the closed subsets {calfl ~ ~ } .
Proof. It follows immediately from (4.1) and (4.2) that d f ( x ) - - 0 if and only if x ~ Ca for some fl ~ ~ . Moreover/~(ca) - ~ is the Stab f-orbit of fl in xe. But if fll and fiE are distinct elements of g+ then their Stab ~-orbits are disjoint. Therefore the subsets {calfl ~ ~ } are disjoint. 4.4. Example. Consider the case discussed in section 3 where X = ($2)" and K = SO(3). We identify with the unit sphere in R 3 and we identify xe and xe* with R 3. Then/~: X~x¢* is given by ~ ( X 1. . . .
S2
,Xn)'~- Ekjxj.
We assume that ~ = (0, 0, II~ll)- The subgroup T of K consisting of rotations about the third coordinate axis in R 3 is a maximal toms. Moreover, its Lie algebra e' is the third coordinate axis in R 3, so t contains ~. The fixed point set of T is the set of all points of the form x s = (xSa..... x s) where S is a subset of {1 . . . . . n} and
xS=
(0,0,1), (0,0,-1),
if i ~ S , ifi~S.
If we identify g with R via the isomorphism (0,0, x) ~ x which sends ~ to fixed point set under the projection/a T of/~ onto t is
~¢={k(S)ISc
11411then
(1 ..... n}},
where
k ( S ) = E k i - E ki" i~S iq~S Thus the set ~ r defined just before (4.2) is
~ r = (0} u ( k ( S ) - II~lllS_c {1 . . . . . n}}. If ~ q: 0 then Stab ~ = T and ~. = ~' is a positive Weyl chamber for Stab ~ in g. Hence ~ = ~TIf ~ = 0 then Stab ~ = K and we may take g+ to be the non-negative real axis. Then ~ = ~i~rn g+= {0} u { k ( S ) -II~lllS -
{1 . . . . .
n }, k ( S ) >
II~ll}.
the image of this
F. Kirwan / Motion of vortices on a sphere
If 0 * f l ~
117
then To= T and
za=zan~,-~(#+~)--~ {xSlSc_(1 . . . . .
n}, k ( S ) - - / 3 + I1~11}.
Clearly the connected components of Z 0 are single points. When ~ * 0 so that Stab ~ = T then Ca = Z a and hence the non-minimal critical points of f are exactly the points of the form x s with S___ { 1 , . . . , n } . When ~ = 0 then S t a b ~ = K and so by (4.2) a point ( x l , . . . , xn) of X is critical for f if and only if there exists a subset S of {1 . . . . . n } and points p, q of S 2 such that
Xim
p,
if i ~ S ,
q,
if iq~S.
Since bt is a K-equivariant map, if x~Zof"lft-l(fl+~) and k ~ Stab~ then k x ~ Z a n p - l ( f l + ~ ) if and only if k ~ Stab (fl, ~) where Stab (fl, ~) = StabB n Stab 4. Therefore Ca --- Stabfl X Stab(a,oZa n # - l ( f l +/~) (cf. [9] 4.2) and hence by [5] section 13 its equivariant cohomology satisfies
=
(zon
+ *))
(4.5)
(for the elementary properties of equivariant cohomology see e.g. [5] section 1). Since X is compact we also know from [9] 5.8 that
H~t~b,( X) -- H*( X) ® n * ( s ( S t a b ~)),
(4.6)
where B(Stab ~) is the universal classifying space for the compact group Stab ~. Our next aim is to show that although f is not a Morse function in the classical sense, nonetheless it is a "minimally degenerate Morse function" and thus gives rise to Morse inequalities (see [9] section 10). For this it is necessary to find for each fl ~ ~ a "minimising manifold" Z 0 f o r f along Ca. That is, Z a must be a closed submanifold containing Co of some neighbourhood of Co in X. Moreover the set where the restriction of f to Z a takes its minimum value must be precisely Ca, and for every x ~ Co the subspace TxZa of TxX must be maximal among all subspaces of Tx X on which the Hessian of f is non-negative. In fact if we fix a Riemannian metric on X then it suffices to find a closed submanifold R a of some neighbourhood of Ca in X with the following properties. R a must contain Ca, the restriction of f to R a must take its minimum value precisely on Ca, for each x ~ Ca the tangent space TxRa should be invariant under the Hessian H of f at x regarded as a self-adjoint endomorphism of TxX via the metric, and finally the restriction of H to the orthogonal complement to TxR a in TxX should be nondegenerate. Then the orthogonal complement U to the restriction of TX to R 0 splits near Ca as U+* U- where H is positive definite on U + and negative definite on U-. It is easy to check that the image Z 0 of a small neighbourhood of Ca in U + under the exponential map Exp: TX---, X is a minimising manifold for f along C0, with codim ~a = codlin R a - dim US for any x ~ C~.
(4.7)
F. Kirwan / Motion of vortices on a sphere
118
Consider any fl ~ ~ . A trivial modification of the proof of [9] 4.11 and 4.15 shows that the intersection of some neighbourhood W of C~ in X with (Stab ~)Z~ is a smooth submanifold of W of dimension dim Zo + dim Stab ~/Stab (13, ~)
(4.8)
and the restriction of f to (Stab ~)Za takes its minimum precisely on Ca. We may choose a K-invariant Riemannian metric on X so that X has a K-invariant almost-complex structure i satisfying %()), f ) = 7).if and i~).i~ = ) ) - ~ for all x ~ X and ~1,~ ~ TxX. (Use e.g. [9] 4.1 to find a metric and almost-complex structure satisfying the first condition, and then replace the metric by 101" S"+ i)) • if).) Then grad f ( x ) = 2i(/z(x) - ~)x for all x ~ X (see the proof of (4.1)). Thus the Hessian H of f at any x e Za n / z - l ( f l + ~) is identified via the metric with the self-adjoint endomorphism H of TxX given as follows. If ~/~ TxX then ½H(Ti) = it(7/) + id/~(x)(~)~,
(4.9)
where *l + fl(~) denotes the action of fl on T~X induced from the action of K on X and the fact that fix = 0. If a ~ xe then d / ~ ( x ) ( a x ) = [a,/z(x)] = [a, 13 + ~1, SO
½H(ax) = i[fl, a]x + idlz(x)(ax) x = - i [ ~ , a] ~.
(4.10)
This vanishes when a ~ stab ~, as it must because f is Stab ~-invariant. Note that since fly = 0 for all y ~ Za and g is an equivariant map, its derivative d/z(x) at x maps TxZa into stabfl. As Za is a Stabfl-invariant almost-complex submanifold of X ([9] 4.12) it follows that H: TxX ~ T~X preserves T~Za and hence also Tx(Stab ~)Za, 4.11. Lemma. The restriction of H to the orthogonal complement (Tx(Stab ~)Z#) ± to Tx(Stab~)Z ~ in TxX is nondegenerate.
Proof. By (4.10) ½H(ax)= - i [ ~ , alx
F. Kirwan/ Motion of vorticeson a sphere
119
if a ~ g and by (4.9) we have ½H(ia~) = - [ f l , a ] x + i A ~ ( a ) x ,
(4.12)
where A~: g ~ g is defined by Ax( a ) . b = d/L(x)(iax)* b = ia~. ib~ = a~* bx
for all b ~ g. Thus the subspace gx + ig~ of T~X is H-invariant. If ~ ~ TxX is orthogonal to this subspace then d~(x)(~)'b=~'i~=O
for all b ~ g, so d/t(x)(,/) = 0. This implies that ½H(•) = ifl(,/) by (4.9). But the endomorphism n ~ fl(*/) of T~X vanishes on T~Z# and is nondegenerate on its orthogonal complement (T~Za).L, which contains (Tx(Stab~)Z#) ± . Therefore since TxZa is a complex subspace of T~X it suffices to show that the restriction of H to (T~(Stab ~)Z~) ± N (gx + igx) is nondegenerate. We now use the simple fact that the kernel of H is the same as the kernel of 1ill. It follows from (4.10) and (4.12) that via the obvious projection gO g--, gx + igx the restriction of ½ill to gx + igx lifts to the linear map E: g ~ g ~ g ~ g represented by the matrix
)
Ax -ad(fl)
where ad denotes the infinitesimal adjoint action of g on itself. By putting E into canonical form we find that the kernel of H restricted to gx + igx is spanned by vectors of the form a~ + ibx with
for some p > 1. Suppose that
=(:) where c ~ s t a b ~ + stabfl and d ~ s t a b f l . Since adfl and ad~ commute and are skew-adjoint we may simultaneously diagonalise them over C and deduce that ker (ad~)2(ad fl) = ker (ad ~)(ad fl) = ker (ad ~) + ker (ad fl) = stab ~ + stab fl
120
F. Kirwan/ Motion of vortices on a sphere
and ker (ad fl )2 = ker (ad r ) = stab ft.
Since (ad fl)2b = - ( a d f l ) d = 0 it follows that b ~ stabfl, and thus (ad fl)(ad ~)2a = (ad fl)(ad ~)Ax(b ) + (ad fl)(ad ~)c = (ad ~)Ax(ad fl(b)) = 0, since ad fl commutes not only with ad ~ but also with A x because fix = 0 and the metric is K-invariant. Hence a ~ stab ~ + stab ft. Therefore by induction if
for some p > 1 then a ~ stab~ + stabfl and b ~ stabfl. This shows that if I-I(a x + ibx) = 0 then a~ + ib x ~ (stabfl)x + (stab ~)~ + i(stabfl) x ~ T~(Stab ~) Z¢ and hence completes the proof of the lemma.
[]
Thus we have proved 4.13. Corollary. f is a minimally degenerate Morse function on X. Clearly f is Stab t~-invariant, and we can take the minimising manifolds ,~¢ to be Stab ~-invariant as well. Hence by [9] (10.2) f induces Stab ~-equivariant Morse inequalities relating the equivariant Betti numbers of X to those of the critical sets C¢. The proof of [9] 5.4, trivially modified, shows that f is equivariantly perfect, i.e. these equivariant Morse inequalities are in fact equalities. Thus using (4.5) and (4.6) we have 4.14. Theorem. The Stab l~-equivariant Betti numbers of C O=/~-1(~) are given by the formula
etStab~(l~-l(~))=Pt(x)et(B(Stab~))
-
E tX(a)etStab(B")(ZaN#-l(fl+ O*B~g~
where X(,8) = codim "~a is the index of f along Ca.
~))
[]
4.15. Remark. The formula of (4.14) makes sense only when the index of f along every component of Ca in X is the same. If Z a has components za, 1 Za, q, say, then we must interpret tx(a)PtStab(a'*)(Z a C~ /~-l(fl + ~)) as .....
E th(B'~)ptStab(B'~)(ZB,jNl't-l(fl+ l
C~,j = Stab I}(Z~ n tt-'(fl + ~));
~)),
F. Kirwan / Motion of vortices on a sphere
121
that is, the number i n d ( H ) of strictly negative eigenvalues of the Hessian H of f at any point x ~ CI~,j. Note that the proof of (4.11) shows that ind ( H ) = ind (i/3l (,x + i~¢.)-L) + ind (HI,x + i,x) = ind (ifl) - ind (C) + ind (Clk~re) + ind ( D ) - ind (Diker(p)), where C and D: x¢O xe+ ~O ~ are the linear maps represented by the matrices 0
-ad(fl) )
ad(fl)
0
- a d ( f l ) ),
-ad(~)
0
A~
which project to the linear maps ifl and ½H: ~¢x+ ilx--+SCx+ ix¢x via the obvious projection p: xeO ,¢--+ ,¢ + ixe. 4.16. Example. Consider again the case discussed in section 3 when X = ( S 2 ) n and K = SO(3). We can use (4.15) to compute the index k(S) of f at the point x s = (xSt. . . . . x s) where S is a subset of {1 .... , n } and (0,0,1), xS= ~ (0,0,-1),
if i ~ S , ifi*S.
Recall from (4.4) that
2 = {0} u
(k(S)-II~llISC_ {1 .....
n}}
if ~ :# 0 whereas 2 = (0} U { k ( S ) [ k ( S ) > 0 } if ~ = 0, and if 0 ~: fl ~ ~ then
Ca = Zan g-x(~+ ~)=Za--
{xSlk(S)=/~+11~11}.
To compute the index X(S) of f at x s we use the formula k ( S ) --- ind Off) - i n d (C) + ind (Clkerp) + ind ( D ) - ind (Diker(p)) of (4.15). If fl = k ( S ) - ~ is positive (when identified with a real number via the isomorphism (0, 0, x) then it is easy to check that i n d ( i f l ) = 2c(S) where c ( S ) is the cardinality of the set { j ~ S l k j > ( j ¢~ S l k j < 0}. If fl = k ( S ) - ~ is negative then ind(ifl) = 2(n - c(S)). We always have ~ = rfl for real number r, so ad(fl), ad(~) and A x all commute and thus may be simultaneously diagonalised. the eigenvalues of
-ad(~)
Ax
~ x) 0} u some Thus
F. Kirwan / Motion of vortices on a sphere
122
are the solutions h to the equations X ( X - g) + r v 2 = 0 , where (#, iv, irv) are simultaneous eigenvalues for (Ax, ad(fl), ad(~)). Hence ~ = ½(# + ~bt2- 4rv2). Since A, is always non-negative definite we have/~ > 0, so one of these values of X is always non-negative. The other is strictly negative if and only if r < 0 and v ~ 0. By assumption ~ > 0. If ~ = 0 then r = 0 and /3 > 0, while if ~ > 0 then r < 0 if and only if/3 < 0. Thus if/3 > 0 then i n d ( D ) = 0, while if/3 < 0 then ind ( D ) = dim K/Stab~3 = dim K / T = 2. A similar but easier argument shows that ind ( C ) = 2 in all cases. It is not hard to check that the index of the restriction of D to k e r ( p ) is the same as that of C in all cases, so we have proved
h(S)=
2 c ( S ) - 2,
if/3 = k ( S ) - I1~11> 0,
2(n-c(S)),
if/3=k(S)-II~ll
(4.17)
Note that if 0 :~/3 ~ ~ then Stab (/3, ~) = Stab/3 = T, so that
etStab(a'O(ZB N/x-l(/3 + ~)) = P(BT) = (1 - t 2) -1 Thus (4.4), (4.14), (4.15) and (4.17) together complete the proof of theorems 3.9 and 3.11. 4.18. Remark. We have already noted that Z~ is a Stab ~-invariant symplectic submanifold of X. Since /3, = 0 for every x ~ Z~ and /x is K-equivariant, the restriction of # to Z~ maps Z~ to the Lie algebra Stabfl of Stab/3 and is a momentum map for the action of Stabfl on Z a (cf. [9] 4.9). Therefore /x - / 3 : Za ---, Stab/3 = (Stab/3)* is also a momentum map for this action. Clearly Za (~/,-1(/3 + ~) is the inverse image of $ under this momentum map. Thus (4.14) gives us an inductive formula for the Stab ~-equivariant Betti numbers of #-1(~) (at least if bt-l(~) is not empty). Of course in the case discussed in section 3 when X= (S2)" and K = SO(3) the Poincar6 series
F. Kirwan/ Motion of vortices on a sphere
123
is easy to c o m p u t e directly, so we only need one step of the induction. I n general we m a y need several steps. T h e inductive formula given by (4.14) can be made into an explicit formula (cf. [9] 5.16) involving only the Betti n u m b e r s of X and certain fixed point sets of torus actions such as Za, together with the Betti n u m b e r s of the classifying spaces of Stab ~ and certain subgroups of Stab ~ such as Stab(fl, ~). W e have n o w shown how to c o m p u t e Ptst~b ~(#-x(~)). Finally in order to relate this to the Betti numbers of the M a r s d e n - W e i n s t e i n reductions X~ we need 4.19. Proposition. Suppose that ~ is a regular value of/x. Then the stabiliser of every x ~ # - t ( ~ ) is finite a n d the Betti numbers of the reduction X~ = / ~ - t ( ~ ) / S t a b ~ are given by
P t ( X ~ ) ~-- PtStab~(]£-l(~)) a n d thus can be c o m p u t e d using (4.14) and (4.18).
Proof. T h e first statement is an immediate consequence of the definition of a m o m e n t u m map: if a ~ xe, x ~ X and 71 ~ Tx X then d~(x)(~)
•a
=
¢Ox(~, a~)
so d # ( x ) is surjective if and only if a x 4= 0 whenever a * 0. The second statement now follows from a well-known p r o p e r t y of rational equivariant c o h o m o l o g y (cf. [9] 5.6). []
References [1] R. Abraham and J.E. Marsden, Foundations of Mechanics (Benjamin-Cummings, New York, 1978). [2] H. Aref, Integrable, chaotic and turbulent vortex motion in two-dimensional flows, Ann. Rev. Fluid Mech. 15 (1983) 345-389. [3] V.I. Amol'd, M~thodes math6matiques de la m6chanique classique, Editions de Moscou (1976). [4] M.F. Atiyah, Convexity and commuting HamJltonians, Bull. London Math. Soc. 14 (1982) 1-15. [5] M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. London A 308 (1982) 523-615. [6] V.A. Bogomolov, Two-dimensional fluid dynamics on a sphere, Itv. Atmos. Ocean. Phys. 15 (1982) 18-22. [7] L.J. Campbell and J.B. Kadtke, Stationary configurations of point vortices, Phys. Rev. Lett. 58 (1987) 670-673. [8] V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982) 491-513. [9] F.C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Princeton Math. Notes 31 (1984). [10] J. Koiller, Note on coupled motions of vortices and rigid bodies, to appear in Phys. Len. A. [111 J. Koiller, S. Pinto de Carvalho, R. Rodriques da Silva and L.C. Concalves de Oliveira, On Aref's vortex motions with a symmetry center, Physica 16D (1985) 27-61. [12] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974) 121-130. [13] J.W. Milnor, Morse theory, Annals of Math. Studies 51 (Princeton Univ. Press, Princeton, 1962). [14] F. Pacella, Central configurations of the N-body problem via equivariant Morse theory, Arch. Rat. Mech. Anal. 97 (1987) 59-74. [15] E.H. Spanier, Algebraic Topology (Tara McGraw-Hill, Bombay, 1971). [16] E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661-692.
T H E T O P O L O G Y OF REDUCED P H A S E SPACES OF T H E M O T I O N OF VORTICES ON A S P H E R E Frances K I R W A N Mathematical Institute, 24-29 St. Giles, Oxford, UK
Received 10 February 1986 Revised manuscript received16 November 1987
The motion of point vortices in a perfect incompressible fluid on a two-dimensional sphere can be represented by a Hamiltonian flowwhich is invariant under the action of the special orthogonalgroup SO(3). Followingthe reduction method of Maxsden and Weinstein this motion can be described by Hamiltonian flows on reduced phase spaces. Some topological invariants of these reduced phase spaces are calculated. For suitable vorticitiesthe results obtained can be used to give lower bounds on numbers of relativeequilibria of the original system. Introduction In [6] Bogomolov models the motion of n cyclones on the earth as a Hamiltonian flow on a phase space which is the product of n copies of a two-dimensional sphere S 2 with a symplectic form depending on the vorticities of the cyclones. (This simple model depends on the assumption that large scale motions in geophysical fluid dynamics are approximately two-dimensional and disregards the earth's rotation.) Both the symplectic structure and the Hamiltonian are invariant under an action of the special orthogonal group SO(3) and hence there is a momentum map /~ whose values are invariants of the flow. Thus following Marsden and Weinstein [12] the motion can also be described for generic values of the vorticities by the associated Hamiitonian flow on a reduced phase space. The aim of this paper is to study the topology of these reduced phase spaces by computing some topological invariants, their Betti numbers (see (3.9) and (3.11)). Knowledge of these gives one some control over, for example, the number of equihbrium points of the reduced Hamiltonian flow when the reduced Hamiltonian is a Morse function. This is also the number of orbits in the original phase space consisting of stationary or uniformly rotating configurations of vortices with a given value ~ ~ R 3 of the invariant function #. We shall obtain a lower bound for this number, which depends on ~ and the vorticities (see theorem 3.16). Of course for some vectors ~ in R 3 there are no configurations at all, uniformly rotating or otherwise, at which # takes the value ~, and in these cases the lower bound is necessarily zero. On the other hand, we shall see that for suitable vorticities and suitable ~ the lower bound becomes large rapidly as the number of vorticities becomes large. The layout of the paper is as follows. Section 1 describes Bogomolov's model and the reduced phase spaces in more detail. Section 2 briefly recalls the definition of the Betti numbers of a manifold and why they are related to the numbers of fixed points of Hamiltonian flows. Section 3 gives the main results and an outline of their proof. Section 4 consists of general nonsense: it gives a general procedure for computing the Betti numbers of reduced phase spaces which is used to prove the assertions made in section 3. I would like to thank J. Koiller for suggesting this problem to me (see also [11] section 12) and for helpful advice. He also pointed out to me that a similar approach to a different problem is taken by PaceUa in [14]. 0167-2789/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
100
F. Kirwan / Motion of vortices on a sphere
1. Point vortices on a sphere
The motion of n point vortices on a two-dimensional sphere flow on the phase space x=s
S 2 can
be represented by a Hamiltonian
2 x - - - x S 2 = ( s 2 ) ".
For simplicity we assume that S 2 is the unit sphere in R 3 The symplectic form on X is
to=
E
kjto ,
l <_j<_n
where % is the pullback to X via the j t h projection onto S 2 of the standard symplectic form on S 2, and kj is the vorticity of the j t h vortex. The Hamiltonian function H is defined as follows. If xj ~ S 2 _ ~3 for 1 < j < n then
H(xi,...,x,,)=-
E
kikjlog(1-x~'xj),
l <_i
where xi • x j is the standard inner product of the unit vectors x i and xj in R 3. (See [5]. For a survey article on vortices see [2].) 1.1. Remark. T h e Hamiltonian function H blows up along the diagonals of X (i.e. when x i = xj for some i ~ j ) . For the time being we shall ignore this problem in order to work as long as possible with compact spaces without boundary. Eventually we shall remove the diagonals (see section 3). The precise form of the Hamiltonian will not in fact matter greatly to us. Its only important property is that it is invariant under the action of the special orthogonal group SO(3) on X. Let the special orthogonal group SO(3) be denoted by K. This group acts naturally on S 2 (as a group of rotations) and hence also on X. It is easy to check that both the symplectic form to and the Hamiltonian function H are invariant under the action of K on X. Let / be the Lie algebra of K and let xe* be its dual space. (Recall that xe consists of all skew-symmetric 3 × 3 real matrices.) The Killing form defines an inner product on xe which is invariant under the action of K on ,( by conjugation. We can identify both / and 1" with R 3 using an orthonormal basis; under these identifications the action of K on / and / * given by conjugation becomes the usual action of K on R 3. Because the symplectic structure on X is preserved by K there is a momentum mapping g: X - , / *
= R3
determined by the following two properties: 1.2. (i) # ( k x ) = k g ( x ) for all x ~ X and k ~ K. (Here k x is the result of acting on x ~ X by k ~ K and k g ( x ) is the result of acting on g ( x ) ~ / * = R 3 by k ~ K). (ii) Given any a ~ / there is a function ga: X--, R which sends x ~ X to the natural pairing g ( x ) • a of g ( x ) ~ g* with a ~ xe. There is also a vector field x ~ ax on X defined by the infinitesimal action of K on
F. Kirwan / Motion of vortices on a sphere
101
X, i.e. d ax = -dT ( exp ( ta ) x ) t_ o. This vector field is the Hamiitonian vector field of the function/£a" That is, d # a ( x ; ~/) = tox(71, a~)
(1.3)
for all x ~ X, ~ ~ TxS and a ~ ~¢. (See [1] 4.2 or [9] section 2 or [12] for more details on momentum mappings.) In our case the momentum mapping St/,: X---- ( s 2 ) n " ~ g f* = R 3
is given by
,(xl,...,x.)= E kj j, l
when ~¢* is identified with R 3 and S 2 is identified with the unit sphere in R 3 (see [2] p. 322). The Hamiltonian function H is invariant under the action of K = SO(3) so d H ( x ; ax) = 0 for all x ~ X and a ~ A¢. By (1.3) this is equivalent to saying that the momentum mapping g is constant along the flow of the Hamiltonian vector field of H, i.e. the flow preserves #-l(/j) for all/j ~ k* = R 3. If ~ ~ xe* = R 3 then #-1(~) is invariant under the subgroup S t a b / j = { k ~ KIk/j = /j } of K. It is easy to check that when/j is nonzero Stab/j is isomorphic to the circle group S x (when xe* is identified with R 3 then Stab/j is the group of rotations about the axis in R 3 defined by/j) whereas when /j = 0 then Stab/j is the whole group K = SO(3). The reduced phase space, or Marsden-Weinstein reduction, associated to any element /j of xe* is the quotient space Xt~= / x - l(/j)/Stab/j = t t - I ( K / J ) / K , where K/J is the orbit of/j (see [12]). If Stab/j acts freely on/~-l(/j) then/j is a regular value of # and X~ is a compact manifold of dimension dim X~ = dim X - dim K - dim Stab/j. The property (1.3) of a momentum map implies that when restricted to #-l(/j) the symplectic form to becomes degenerate precisely along the directions tangential to Stab/j orbits. Thus to induces a symplectic form to~ on the reduced phase space X¢ (provided that/j is a regular value of/~). Since the function H is K-invariant it induces a function He on X¢ and the Hamiltonian flow of H~ lifts to the Hamiltonian flow
102
F.
Kirwan / Motion of vortices on a sphere
of H restricted to F-I(~). In particular the equilibrium points for the Hamiltonian flow of H~ on X~ correspond to those solution curves in #-1(~) of the original system which are orbits of one-parameter subgroups of K (see [3] appendix 5), i.e. to configurations of vortices which either rotate uniformly about some axis in R 3 or are stationary. The aim of this paper is to compute the Betti numbers of the reduced phase space X~ when ~ is a regular value of # and to use these to obtain information on the number of equilibrium points for the Hamiltonian flow of H~ on X~. (Since H blows up along the diagonals in X, we shall actually be interested in equilibrium points not contained in the diagonals.) The Betti numbers of a manifold are topological invariants; in the next section we recall their definition and how they are related to the number of equilibrium points of a Hamiltonian flow. For other examples of the application of Marsden-Weinstein reduction to problems involving vortices, see e.g. [10, 11].
2. Betti numbers and Morse theory We recall the definition of the Betti numbers
b o ( Y ) , b l ( r ) , b2(Y) . . . . of a topological space Y. These are nonnegative integers which contain information about the topology of Y. A singular i-simplex in Y is a continuous map a: A t ~ Y where A i is the convex hull of the i + 1 points (0 . . . . . 0 ) , ( 1 , 0 . . . . . 0),(0,1,0 . . . . . 0) . . . . . (0 . . . . . 0,1) in R i. The vector space Ci(Y; R) of singular /-chains with real coefficients consists of all formal linear combinations
ala t + . . . + apav of singular i-simplices with real coefficients aj ~ R. A singular ( i - 1)-simplex is a face of a singular /-simplex a if it is the composition of a: A i ~ Y with one of i + 1 standard maps identifying Ai_ 1 with the faces of A r The boundary of a is defined by
aa=
~
+~,
1" face o f o
where the signs depend on a suitable choice of orientations. We can extend O by linearity to a map a: ci(Y; R) --, C,_l(Y; R ). Then a 2= 0. The (singular) homology groups of Y with real coefficients are now defined to be the quotient vector spaces Hi(y)=
( *1e C / ( Y ; R ) I 0 n = 0}
F. Kirwan / Motion of vortices on a sphere
103
If Y is a reasonably well-behaved space (e.g. a manifold) then these vector spaces Hi(Y ) are finitedimensional and the Betti numbers of Y are defined by
bi(Y ) = dim Hi(Y ). For example when Y-- S 2 then 1, 0,
bi(Y) =
i = 0,2, otherwise.
(2.1)
The cohomology groups Hi(y) of Y with real coefficients are the duals of the homology groups Hi(Y ). When Y is a compact manifold they are the same as the De Rham cohomology groups of Y. The information given by the Betti numbers is encoded in the Poincar6 series
Pt(Y) = y' bi(Y)t i i>O
of Y. If Y is a manifold of dimension m then bi(Y ) = 0 if i > m and so we can define the Euler number
x ( Y ) of Y by
(2.2)
x(Y) = ~, ( - 1 ) i b i ( Y ) = P _ l ( Y ) . i>O
It has the nice property that if Z is a submanifold of codimension y in Y then
(2.3) This can be deduced easily from the existence of a long exact sequence
... ~ H i - ~ ( Z ) ~ H i ( Y ) ~ H i ( y - Z ) ~ H i - r + l ( z ) ~
...
(2.4)
called the T h o m - G y s i n sequence. (See e.g. [15] for more details.) Now suppose that Y is a compact manifold and f: Y ~ R is a smooth function. By perturbing f an arbitrarily small amount we may assume that f is a Morse function, i.e. every critical point x ~ Y for f is isolated and nondegenerate in the sense that the Hessian of f at x (which is given in local coordinates by the matrix of second partial derivatives of f at x) has maximal rank. The index i/(x) of f at x is then defined to be the number of strictly negative eigenvalues of the Hessian of f at x. The Betti numbers of Y are related to the number of critical points of f (i.e. the points at which the derivative of f vanishes) by the famous Morse inequalities, which can be expressed most easily in the form
~., t i / ( x ) - P t ( Y ) = R ( t ) ( l + t ) ,
R(t)>O.
(2.5)
x critical
That is, R(t) is some polynomial in t with nonnegatioe integer coefficients. In special cases (when f is a perfect Morse function) R(t) = 0 so the Betti numbers bi(Y ) are equal to the numbers of critical points of f with given index i. In general putting t = - 1 we get
x(Y)=P_x(Y) =
~, x critical
(-1)/:(x),
(2.6)
104
F. Kirwan / Motion of vortices on a sphere
so the total number of critical points is always at least the modulus
Ix(Y)l
(2.7)
of the Euler number of Y. The Morse inequalities can be derived by putting a Riemannian metric on Y and looking at the induced gradient flow of the function f on Y. The path of steepest descent of any point of Y converges to a critical point for f , and we can stratify Y by putting points into the same stratum if and only if their paths of steepest descent converge to the same critical point. The codimension of the stratum corresponding to a critical point x is the index if(x). By applying the T h o m - G y s i n sequence to each stratum in turn we obtain the Morse inequalities. For alternative arguments see [13] and [16]. If the critical points of f are not all isolated then the total number of critical points is infinite and so trivially greater than Ix(Y)I. However, if the critical points form submanifolds of Y and the Hessian is nondegenerate in directions normal to these submanifolds, then we can obtain more sophisticated Morse inequalities
~,ti:(C)pt(c ) - Pt(Y) = R ( t ) ( 1 + t),
R ( t ) > O,
where the sum runs over all the connected components C of the set of critical points and if(C) is the index of f at any point of C. If Y is not compact, then the Morse inequalities may fail (for example, simply remove all the critical points from Y). However, if Y is a compact Riemannian manifold and Z is a closed subset and f: Y - Z ~ R is a Morse function such that the path of steepest descent of every point of Y - Z converges to a point of Y - Z, not Z, then the proof of the Morse inequalities still holds and hence the total number of critical points of f is at least
Ix(r- z)I.
(2.8)
In particular, these conditions are satisfied if f ( y ) ~ oo whenever y ~ Y - Z approaches Z. The Morse inequalities together with (2.3) can also be used to show that if f: Y - Z ~ R attains its minimum value on a closed submanifold W of Y - Z then the total number of critical points of f in Y - Z - W is at least Ix(Y-
z) - x(W)[ =
Ix(Y- z - w)I.
(2.9)
We can apply this to study the number of critical points of the Hamiltonlan function H~ defined in section 1 (or equivalently the number of equilibrium points of the Hamiltonian flow). Let A = ( ( x x. . . . . xn) ~ X = (S2)nlXl = xj for some i :~j ) and
AS = (A N #-l(~))/Stab ~ c X~. Let X ° -- X - A and X~° = X~ - A~. Then H: X ° -) R is defined by
H ( x 1. . . . . x . ) = -
• l
kikjlog(1-x,.xj)
F. Kirwan / Motion of vortices on a sphere
105
and induces H,: X~° ~ R. Let A += ( ( x t ..... xn) ~ XIx i = xj, some i * j such that kik j > 0} and
a [ = (a+n # - t ( ~ ) ) / S t a b ~ ___X~, Then the function exp H: X - A + ~ R defined by exp H ( x 1. . . . . x , ) =
1-I
(1-xi'xj)
\ -kikj
l
is a smooth function invariant under the action of K--SO(3). Hence it induces a function expH~: X ~ - A T ~ R which takes its minimum value 0 precisely on A ~ - za~, and its other critical points are exactly the critical points of H~: X~° --, R. Moreover it is easy to check that a path of steepest descent for exp H from any point of X~ - A~- cannot converge to za~- and hence its limit points lie in X~ - A~-. Since
the argument for (2.9) shows that provided the values k i of the vorticities are such that the Hamiltonian function H~ is a Morse function on X,°, the number of its equilibrium points is at least
Ix(x,0)l
(2.10)
In the next section we shall see how this Euler number x ( X ~ ) can be calculated using equivariant Morse theory. So to complete this section we need a few remarks about equivariant cohomology and equivariant Morse theory. For this we assume that a compact group K acts on Y. Let
E K --* B K be the universal classifying bundle of K. Then EK is a contractible space on which K acts freely and BK is the quotient of E K by K. We define the equivariant cohomology groups H~c(Y ) of Y to be the ordinary cohomology groups of the quotient
Y ×r E K of Y × E K by the diagonal action of K. The point of this is that if K acts freely on Y (or more generally if every x ~ Y is fixed by only finitely many elements of K ) then
Hk(Y) = Hi(y/K),
(2.11)
but in general H~c(Y) has better properties than H i ( y / K ) . If f: Y ~ R is a K-invariant Morse function we can do equivariant Morse theory replacing ordinary cohomology by equivariant cohomology throughout, and we obtain equivariant Morse inequalities relating
F. Kirwan ~Motion of vortices on a sphere
106
the equivariant Poincar6 series
Ptk(Y) = E tidimHk(Y) i>O
of Y to the equivariant Poincar6 series and indices of the connected components of the set of critical points for f.
3. The results
We saw in section 1 that the momentum mapping #: X ~ / * X = ($2) n with symplectic form
Ekj
l
= R 3 for the action of K = SO(3) on
j
is given by
~(x 1..... x,)=
E ~xj. l
We are interested in computing the Betti numbers of the reduced phase space X¢ = # - l ( ~ ) / S t a b when ~ is a regular value of #, in order to obtain bounds for the number of fixed points of Hamiltonian flows on X~ (see section 2). The idea is to apply equivariant Morse theory to the function f : X--,R defined by f(xl .....
x=) - 6 II=.
Here II II denotes the usual length function on R 3 which is invariant under the action of K = SO(3), so f is invariant under the action of Stab ~. It turns out (see (4.14)) that the function f is equivariantly perfect in the sense that its equivariant Morse inequalities are actually equalities. In other words the equivariant Poincar6 series of X with respect to the action of Stab ~ satisfies PtStab~( x ) = Eti'(c)PtStab~( c ) ,
(3.1)
where the sum is over all connected components C of the set of critical points for f, and if(C) denotes the index of f along C. (That is, i/(C) is the number of strictly negative eigenvalues of the Hessian of f on the tangent space to X at any x ~ C.)
F. Kirwan/ Motion of vortices on a sphere
107
It is clear that f attains its minimum value precisely on the subset #-1(4) of X. Therefore we can rearrange (3.1) to get
PtSUa'~(#-a(4))=PtStab~(x) -
E
tq(C)p,S'~'~(C) •
(3.2)
c¢~,-1(~) By assumption 4 is a regular value of #, i.e. if x ~ #-1(4) then the derivative of # at x has maximal rank. Because of the defining property (1.3) of a momentum map this means that any x ~ / ~ - l ( x ) is fixed by only finitely many elements of Stab 4. Therefore by (2.11) the Poincar~ polynomial of X~ = # - 1 ( 4 ) / S t a b 4
satisfies Pt(X~) = P t S ~ ( # - l ( 4 ) )
(3.3)
and hence is given by the right-hand side of (3.2). On the other hand, because X is a compact symplectic manifold and the group action preserves its symplectic structure we have PtStab£(X) = Pt( X)Pt(B(Stab 4)),
(3.4)
where B(Stab 4) is the classifying space of Stab 4 (see [9] 5.8). If 4 is nonzero then Stab ~ is isomorphic to the circle group S t and
Pt(B(Stab 4)) = Pt( B $1) = 1 + t 2 + t ' + . . . .
(1 - t 2)-1.
(See e.g. [5].) If 4 = 0 then Stab 4 = S0(3) and
Pt(B(Stab 4)) = Pt(B S0(3)) = 1 + t ' + t 8 + . . . .
(1 - t ' ) - l .
Moreover
Pt(X)=(Pt(S2))"=(I
+t2)"
Thus ptSt~b~(x) =
( 1 + t 2 ) " ( 1 - - t 2 ) -1, ( 1 + t 2 ) " - l ( 1 - - t 2 ) -1,
if4*0, if4=0.
(3.5)
Finally we must deal with the terms
tiI(C)PtStab~( c ) in (3.2). Note first that if 4 and [ lie in the same K-orbit in ~¢* = R 3 then the reduced phase spaces X~ and
F. Kirwan / Motion of vortices on a sphere
108
X~ are the same: X~ = # - l ( ~ ) / S t a b ~ = ~ - I ( K ~ ) / K = # - I ( K ~ ) / K = X~. Hence we may assume without loss of generality that ~ is of the form = (0,0, I1~11).
(3.6)
Then it turns out (see (4.4)) that when ~ is nonzero the critical points of f~ not contained in #-1(~) are the points .... ,
x=
satisfying xl -- (0, 0, + 1) for 1 < i _< n. There are exactly 2 n such points, one for each subset S of {1 ..... n }. We write
xS = ( x s .... , x s) where [ (0,0,1), xS= ~ (0,0,-1)
if i ~ S ,
ifi~S.
If ~ = (0,0, [1~11)is nonzero then Stab ~ is the group of rotations about the third coordinate axis in R 3, and hence Stab ~ fixes each point x s of the form above. The equivariant Poincar6 series of any single orbit is always Pt(BY~) where ~ is the stabiliser of a point in the orbit (see e.g. [5] section 13). In this case = Stab ~ and we get
ptStab*( { x S ) ) = Pt( B(Stab ~ ) ) = Pt( BS 1) = (1 - t2) -1. Thus combining (3.2), (3.3) and (3.5), when ~ is nonzero we get
Pt(X~) = (1 + / 2 ) n ( 1 - t2) - 1 -
E
tx(s)(1 - rE) -1,
(3.7)
SC_{l . . . . . . )
where X(S) is the index of f at the point x s. It is shown in section 4 that this index can be computed as follows. If S is a subset of {1,..., n } let k(S):
Ekjj~S
j~S
and
c(S)= #{jESlkj>O
} + #{jq~S[kj
F. Kirwan/ Motionof vorticeson a sphere
109
where # F denotes the number of elements of any finite set F. Then if ~ q: 0 is a regular value for # and S is a subset of {1 . . . . . n} we have k(S) ~ I1~11and
~(s)
[ 2c(S), [ 2(n -
c(S)),
if
k(S)
if
k(S) <
> II~ll,
(3.8)
II~ll-
(See (4.17).) Combining (3.7) and (3.8) we obtain 3.9. Theorem. Let ~ be a nonzero regular value of/~. Then the Poincard polynomial phase space X~ is
(1--t2)-1[ (l + t2)n- k(S)Y'~II~ll Examples. (a) If the vorticities + • • • + k . - 2 rain kj we have
3.10.
Pt(X~) of
the reduced
k(s)Y]~I~l t2(n-c(S))] k I . . . . , k , are all positive, then when k I + . . . + k , > I1~11> kl
k(S) > I1~11"~S = ( 1 , . . . , n }, and hence
Pt(Xt~)=(a-t2)-2[(l+t2)n-t2
2 + t a + "-" + t 2("-2). "" >k,>O
then when k l + . . .
+k,_l-k,>ll~ll>kl + ... + k , _ 2 - k , _ x + k . we
k(S)>ll~ll**S={1,...,n} or {1 .... , n - l } , and then
Pt(X~) = (1
- t2)-111 + t 2 -- t 2 ( n - l ) - - / 2 ( n - 2 ) ]
= 1 + 2 t 2 + 2t 4 + • • • + 2 t 2(n-3) + t 2(n-2). N o t e that as in (a) this is a polynomial of degree d = 2n - 4 in t with nonnegative coefficients and the coefficient of tJ is the same as the coefficient of t a-j. Any compact manifold of dimension d must have a Poincard polynomial of this form, by Poineard duality. We can deal with the ease ~ = 0 similarly. The result is as follows (see section 4 for the proof). 3.11.
Theorem.
If 0 is a regular value of # then the Poincard polynomial
Pt(Xo)
of the reduced phase
110
F. Kirwan / Motion of vortices on a sphere
space X 0 is
(1-- t2)-l[(l + t2)n-l-- ~" t2(c(S)-l)]. k(S)>O
J
Example. If the vorticities k x. . . . , k~ are all positive and sufficiently close together then
3.12.
k(S) > 0 "~ c(S) > n/2 and hence
Pt(Xo) = (1
-/2)-1[(1
"q- / 2 ) n - 1
_
[
n>r>n/2
= 1 +b2t2q - - ' - + b 2 j t 2 J +
...
-[-t 2(n-3),
where
b2j=l+(n_l)+(n-l)+...+(
n-1 ) min(j, n- 3-j) "
2
By theorem 3.9 if ~ is a nonzero regular value o f / t then the Euler number x(X~) of the reduced phase space X~ is x(X
) =
=
=
lim
(~'k(S)'ll~llt2(c(S)-l)q- ~k(S)
t2-_.l
( t 2 - 1)
E
(c(S)-l)+
k(S) > I1~11
E
(n-c(S))-n2
n-'
(3.13)
k(s) < I1~11
By theorem 3.11 when ~ = 0 we get
X( Xo) = P-I( Xo) = lim (Ek(S)>°t2~s)-l~-- (1 + t2) "-1) t2.... 1
= •
( t 2 - 1)
(c(S)- 1)- (n- 1)2n-2.
(3.14)
k(S)>0
We noted at (2.7) that Ix(Se)l is a lower bound for the number of critical points of a real valued Morse function on X~, and hence also for the number of fixed points of a Hamiltonian flow on X~ defined by a Morse function. Unfortunately, the Hamiltonian H e defined in section 1 blows up along the image A~ in X~ of the diagonals of X = ($2) n. We saw at (2.10) that to obtain a lower bound for the number of fixed points in X~° = X~ - A~ of the Hamiltonian flow on X~° induced by H~ we must compute
Ix(x,°tl
F. Kirwan / Motion of vortices on a sphere
111
For this we divide X~ into subsets indexed by equivalence relations on (1 . . . . , n }, or equivalently by partitions of {1 . . . . . n}. (Recall that to an equivalence relation corresponds the partition defined by its equivalence classes.) If ~r is a partition of {1 .... , n } let A~-- {(xx . . . . . x , , ) ~ X [ x i = + ' ~ i - , j } and A'~ = ( a " Cl/~-l ( ~ ) ) / S t a b ~ __ X,, where - , , is the equivalence relation corresponding to or. Note that if ¢r0 denotes the trivial partition {{1}, [2} . . . . . {n}} then
a?= xg, and
~ r ~ c ro
Moreover X~ is the disjoint union of the A~ which are locally closed submanifolds of even codimension in X~. Hence by (2.3) x(X~°)=x(X~) -
E x(A~) •
(3.15)
crier o
The reduced phase space X~ depends on n and the vortices k 1. . . . . k n as well as on £. Let us write
X~( k 1..... k,,),
X~( k 1..... k,,)
for X~ and X~° when it is necessary to demonstrate the dependence on n and k 1. . . . . k. explicitly. Then if
rr= { A~,..., Aq } is any partition of {1 . . . . . n } into disjoint nonempty subsets A~,...,
Aq it is easy to see that
A'~ _---x~o(kl, . . . . , k~(,,)), where n (~r) = q and
ky=Ek,. i~Aj
Thus (3.15) gives an inductive formula for X(X~°) in terms of X(X~) (which we have already computed: see (3.13) and (3.14)) together with terms of the same form as x(X~°) but with smaller values of n.
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F. Kirwan/ Motion of vorticeson a sphere
3.16. Theorem. If the vorticities k i are such that the reduced Hamiltonian H~ is a Morse function then the number of equilibrium points of the Hamiltonian flow of H~ on X~° is at least
where the Euler number X (X~°) = X( X~°(kl . . . . . k,)) is given inductively by the formulas
x ( g : ( k l .... ,kn)) ~- x( g~(ka ..... kn)) x(X¢(kx,...,k,))=
~.,
(c(S)-
1) +
k(S) > IIEI if ~ 0
E x ( S : ( k ' ( ..... k:(~r,)), ~rq-~ro E
(n-c(S))
-n2"-1
k (S) < I1~11
and
x ( g o ( k l ..... kn)) =
E
( c ( S ) - 1) - ( n -
1)2 n - 2 .
k(S)>0
This theorem gives us a lower bound for the number of stationary or uniformly rotating configurations of vortices on S 2 with a given value f of the momentum mapping. However, the formulas are rather complicated and it is not immediately clear that they can tell us anything interesting, so let us consider some special cases. Of course when f does not lie in the image of/~ then X~ = q~ and the lower bound we get has to be zero. This happens if and only if either (a)
II~ll>lkll+lk21+-.-+lk,I
(3.17)
or there exists some j ~ ( 1 , . . . , n ) such that (b)
II~ll
< Ikjl- E Ikil • i ~j
If equality holds in either (a) or (b) then $ is not a regular value of/~, except in the trivial case n = 1 when the lower bound is
x(Sg) =x(S~)=
( O' 1,
II~n=~:]kll'
II~ll=lkxl.
F r o m now on we shall assume that ~ is nonzero and
Ikjl- ~lkil
< l k a l + "'" +tk.I
for all j. When n > 2 the dimension of S~ is 2n - 4. When n = 2 X~ is a single point (which is automatically an equilibrium point of the flow). This is consistent with the calculations
Pt(Xf) = ( 1 - t2)-1((1 + t 2 ) 2 _ t 2 _ t 2 _ t 2 _ t4) = 1
F. Kirwan ~Motion of vortices on a sphere
113
and
x(X,)-- 1. When n = 3 we find
Pt(X~) = (1 - t2)-1((1 + t2) 3 - t 4 - 3t 2 - 3t 4 - t 6) = 1 + t 2, so
x( X~) = 2 and x(Xg)=2-x(Xg(kl,
k 2 + k 3 ) ) - x ( X g ( k 2 , kl + k 3 ) - x ( X g ( k 3 , kl + k 2 ) ) .
Thus x(X~°) = 2,1,0 or - 1 depending on whether none, one, two or all of the conditions (i)
[ I k l l - Ikz + k311 < I1~11< Ikll + Ik2 + k31,
(ii)
Ilk21- [kx + k3ll < I1~11< Ik21 + Ikx + k31,
(iii)
Ilk31-Ik~ + kzll < I1~11< Ik31+ Ikl + kzl
are satisfied. When n > 3 there are too many different cases to be listed here. However, in the special case when the vortices kl . . . . . k n are all positive and I1~11is strictly less than but close to k 1 + . . . +k n we have already seen at (3.10) that
x( X~) = P_I( X~) = n - 1 . It follows from (3.16) that
X(X~)=h(n), where
h(n) is defined inductively by
h(1)=0,
h(n)=n-1-
~_,
p(n,k)h(k)
l
and p(n, k) is the number of partitions of {1 . . . . . n } into k nonempty subsets. Thus h(2) = 1, h(3)=2-3.1=
-1,
h(4) = 3 - 7 . 1 - 6 . ( - 1 )
=2,
h(5) = 4 - 1 5 . 1 - (10 + 15). ( - 1 ) -
10.2 = - 6 ,
h(6) = 5 - 3 1 . 1 - (15 + 6 0 + 1 5 ) - ( - 1 ) -
(45 + 2 0 ) - 2 - 15-(6) = 89.
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114
This shows the lower bound Ih(n)l on the number of stationary or uniformly rotating configurations increasing rapidly as n increases.
4. General n o n s e n s e
In this section we shall prove the assertions made in the calculation of the Poincard polynomials Pt(X~) in section 3. We shall work in the following more general setting. Let X be a compact symplectic manifold with symplectic form to and let K be a compact connected Lie group acting on X in such a way that the symplectic structure is preserved. Let t be the Lie algebra of K and let t * be its dual. Suppose that t~: X - , t * is a momentum map for the action of K. This means that for any a ~ 1 the component of /t in the direction of a is a Hamiltonian function for the vector field x --, a x on X induced by a; that is d # ( x ) ( ~ ) - 1= tox(~i, a x ) for all x ~ X and ~ ~ TxX; and moreover I ~ ( k x ) = Ad* k l ~ ( X )
for all k ~ K and x ~ X where Ad* is the co-adjoint action of K on l * (see e.g. [12], [1] 4.2 or [9] section 2 for details about momentum maps). Then the reduced phase space, or Marsden-Weinstein reduction, associated to any element ~ of l * is the quotient space x¢ =
where Stab ~ is the stabiliser of ~ under the co-adjoint action of K on 1 * (see [12]). If Stab ~ acts freely on g - i ( ~ ) then ~ is a regular value of # and X~ is a compact manifold of dimension dim X~ = dim X - dim K - dim Stab ~. Let us fix a K-invariant inner product on 1 (this also gives an invariant norm I1 II on l ) and use it to identify l * with t . In [9] equivariant Morse theory is applied to the function IINI2 on S to obtain a formula for the Betti numbers of the reduced phase space X 0 = / ~ - i ( 0 ) / K provided that 0 is a regular value of #. In this section we shall apply similar ideas to the function f = Ib - 6112 to obtain a formula for the Betti numbers of X~ for any ~ ~ t * = 1 which is a regular value of/x. First let us consider the critical points of the function f. For the time being we shall not need to assume that ~ is a regular value of #. Let stab ~ = ( a ~ l l [ a , 6] = 0) be the Lie algebra of Stab ~. If a ~ 1 let x ---, a x denote the vector field on X induced by a. 4.1. L e m m a . d r ( x ) = 0 if and only if # ( x ) = ~ + fl where fl ~ stab ~ and fix = 0.
F. Kirwan/ Motion of vortices on a sphere
115
Proof. The proof of [9] lemma 3.1 shows that d f ( x ) = 0 if and only if (g(x) - f)x = 0. In fact it follows easily from the definition of a momentum map that
d f ( x ) ( * / ) = ~x(7/,2(la(x) - f ) x ) for every x ~ X and 71~ T~X. If .8 = #(x) - f and .8~ = 0 then [/3, g(x)] = 0 because/~ is a K-equivariant map. Hence
0 = [ . 8 , . 8 + f ] = [.8,f], s o ,8 ~ stab f.
[]
Suppose that .8 is an clement of xe. Let Ta be the subtorus of K generated by .8. Then the inner product g ( x ) . . 8 of g ( x ) and .8 is constant on every connected component of the fixed point set FIX(Ta) of Ta in X (see e.g. [9] 3.7). Let Za be the union of those connected components of Fix(T~) on which g ( x ) . / 3 = (.8 + f ) " .8. Then Z~ is a Stab.8-invariant symplectic submanifold of X ([8] 3.6 or [9] 4.8). Let T be a maximal torus of K with lie algebra t such that f ~ ~'. Then the orthogonal projection /~r: X -o , ~ [ * of tt onto d is a momentum mapping for the action of T on X ([9] 3.3). The image under tt r of the fixed point set of T is a finite subset ~¢ of ~'. It has been proved by Atiyah in [4] and by Guillemin and Sternberg in [8] that g T ( X ) = Conv~¢, the convex hull of a¢ in ~. For each nonempty subset ~¢' of a¢, the convex hull Conv ( a - fla ~J¢'
} = Conv
'- f
contains a unique point .8(a¢') which is closest to the origin of ¢' (i.e. at which the norm takes its minimum value). Let ~ r be the finite subset of ¢' consisting of all such points .8(z¢'). Let g = ~ r n ~'+ where ~'+ is a fixed positive Weyl chamber for Stab f in ~. (Note that T is a maximal torus of Stab f as w e l l as K.)
4.2. Lemma. Suppose that d f ( x ) = 0. Then there exists k ~ Stab f and .8 ~ ~ such that g ( k x ) - f = .8. Proof. By (4.1) g ( x ) - f = .8 where .8~ = 0 and .8 ~ stab1. Therefore there exists k ~ Stab1 such that Ad(k).8 ~ ~+. Since g ( k x ) - f = Ad ( k ) g ( x ) - f = Ad (k).8,
we may thus assume that .8 ~ ~+ and it suffices to show that fl ~ ~ . We have x E Z# because .8~ = 0 and tt(x)..8 = (.8 + f ) " .8. Moreover g ( x ) = gT(X) because g ( x ) ~ ~'. Therefore by the theorem of Atiyah and Guillemin-Sternberg ([9] 3.4) applied to the action of T on Za we have .8 = ~ ( x )
- f ~ ~T(z~)
- f = Conv~'
- f,
where ~¢' is the subset gr(Z[~ t3 Fix T ) of ~¢=/.tT(Fix T). Furthermore p-(~T(x)
- f ) = 11.sll 2
F. Kirwan/ Motion of vortices on a sphere
116
for all x ~ Zp, so if 71~ Conv.a¢' - ~ then so fl ~.~'. [] If f l ~
117/11>--IIBII with equality if and
only if n = ft. Hence fl = fl(~¢')
let Ca= S t a b ~ ( Z o n / ~ - l ( f l + ~)).
4.3. Corollary. The set of critical points for f on X is the disjoint union of the closed subsets {calfl ~ ~ } .
Proof. It follows immediately from (4.1) and (4.2) that d f ( x ) - - 0 if and only if x ~ Ca for some fl ~ ~ . Moreover/~(ca) - ~ is the Stab f-orbit of fl in xe. But if fll and fiE are distinct elements of g+ then their Stab ~-orbits are disjoint. Therefore the subsets {calfl ~ ~ } are disjoint. 4.4. Example. Consider the case discussed in section 3 where X = ($2)" and K = SO(3). We identify with the unit sphere in R 3 and we identify xe and xe* with R 3. Then/~: X~x¢* is given by ~ ( X 1. . . .
S2
,Xn)'~- Ekjxj.
We assume that ~ = (0, 0, II~ll)- The subgroup T of K consisting of rotations about the third coordinate axis in R 3 is a maximal toms. Moreover, its Lie algebra e' is the third coordinate axis in R 3, so t contains ~. The fixed point set of T is the set of all points of the form x s = (xSa..... x s) where S is a subset of {1 . . . . . n} and
xS=
(0,0,1), (0,0,-1),
if i ~ S , ifi~S.
If we identify g with R via the isomorphism (0,0, x) ~ x which sends ~ to fixed point set under the projection/a T of/~ onto t is
~¢={k(S)ISc
11411then
(1 ..... n}},
where
k ( S ) = E k i - E ki" i~S iq~S Thus the set ~ r defined just before (4.2) is
~ r = (0} u ( k ( S ) - II~lllS_c {1 . . . . . n}}. If ~ q: 0 then Stab ~ = T and ~. = ~' is a positive Weyl chamber for Stab ~ in g. Hence ~ = ~TIf ~ = 0 then Stab ~ = K and we may take g+ to be the non-negative real axis. Then ~ = ~i~rn g+= {0} u { k ( S ) -II~lllS -
{1 . . . . .
n }, k ( S ) >
II~ll}.
the image of this
F. Kirwan / Motion of vortices on a sphere
If 0 * f l ~
117
then To= T and
za=zan~,-~(#+~)--~ {xSlSc_(1 . . . . .
n}, k ( S ) - - / 3 + I1~11}.
Clearly the connected components of Z 0 are single points. When ~ * 0 so that Stab ~ = T then Ca = Z a and hence the non-minimal critical points of f are exactly the points of the form x s with S___ { 1 , . . . , n } . When ~ = 0 then S t a b ~ = K and so by (4.2) a point ( x l , . . . , xn) of X is critical for f if and only if there exists a subset S of {1 . . . . . n } and points p, q of S 2 such that
Xim
p,
if i ~ S ,
q,
if iq~S.
Since bt is a K-equivariant map, if x~Zof"lft-l(fl+~) and k ~ Stab~ then k x ~ Z a n p - l ( f l + ~ ) if and only if k ~ Stab (fl, ~) where Stab (fl, ~) = StabB n Stab 4. Therefore Ca --- Stabfl X Stab(a,oZa n # - l ( f l +/~) (cf. [9] 4.2) and hence by [5] section 13 its equivariant cohomology satisfies
=
(zon
+ *))
(4.5)
(for the elementary properties of equivariant cohomology see e.g. [5] section 1). Since X is compact we also know from [9] 5.8 that
H~t~b,( X) -- H*( X) ® n * ( s ( S t a b ~)),
(4.6)
where B(Stab ~) is the universal classifying space for the compact group Stab ~. Our next aim is to show that although f is not a Morse function in the classical sense, nonetheless it is a "minimally degenerate Morse function" and thus gives rise to Morse inequalities (see [9] section 10). For this it is necessary to find for each fl ~ ~ a "minimising manifold" Z 0 f o r f along Ca. That is, Z a must be a closed submanifold containing Co of some neighbourhood of Co in X. Moreover the set where the restriction of f to Z a takes its minimum value must be precisely Ca, and for every x ~ Co the subspace TxZa of TxX must be maximal among all subspaces of Tx X on which the Hessian of f is non-negative. In fact if we fix a Riemannian metric on X then it suffices to find a closed submanifold R a of some neighbourhood of Ca in X with the following properties. R a must contain Ca, the restriction of f to R a must take its minimum value precisely on Ca, for each x ~ Ca the tangent space TxRa should be invariant under the Hessian H of f at x regarded as a self-adjoint endomorphism of TxX via the metric, and finally the restriction of H to the orthogonal complement to TxR a in TxX should be nondegenerate. Then the orthogonal complement U to the restriction of TX to R 0 splits near Ca as U+* U- where H is positive definite on U + and negative definite on U-. It is easy to check that the image Z 0 of a small neighbourhood of Ca in U + under the exponential map Exp: TX---, X is a minimising manifold for f along C0, with codim ~a = codlin R a - dim US for any x ~ C~.
(4.7)
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118
Consider any fl ~ ~ . A trivial modification of the proof of [9] 4.11 and 4.15 shows that the intersection of some neighbourhood W of C~ in X with (Stab ~)Z~ is a smooth submanifold of W of dimension dim Zo + dim Stab ~/Stab (13, ~)
(4.8)
and the restriction of f to (Stab ~)Za takes its minimum precisely on Ca. We may choose a K-invariant Riemannian metric on X so that X has a K-invariant almost-complex structure i satisfying %()), f ) = 7).if and i~).i~ = ) ) - ~ for all x ~ X and ~1,~ ~ TxX. (Use e.g. [9] 4.1 to find a metric and almost-complex structure satisfying the first condition, and then replace the metric by 101" S"+ i)) • if).) Then grad f ( x ) = 2i(/z(x) - ~)x for all x ~ X (see the proof of (4.1)). Thus the Hessian H of f at any x e Za n / z - l ( f l + ~) is identified via the metric with the self-adjoint endomorphism H of TxX given as follows. If ~/~ TxX then ½H(Ti) = it(7/) + id/~(x)(~)~,
(4.9)
where *l + fl(~) denotes the action of fl on T~X induced from the action of K on X and the fact that fix = 0. If a ~ xe then d / ~ ( x ) ( a x ) = [a,/z(x)] = [a, 13 + ~1, SO
½H(ax) = i[fl, a]x + idlz(x)(ax) x = - i [ ~ , a] ~.
(4.10)
This vanishes when a ~ stab ~, as it must because f is Stab ~-invariant. Note that since fly = 0 for all y ~ Za and g is an equivariant map, its derivative d/z(x) at x maps TxZa into stabfl. As Za is a Stabfl-invariant almost-complex submanifold of X ([9] 4.12) it follows that H: TxX ~ T~X preserves T~Za and hence also Tx(Stab ~)Za, 4.11. Lemma. The restriction of H to the orthogonal complement (Tx(Stab ~)Z#) ± to Tx(Stab~)Z ~ in TxX is nondegenerate.
Proof. By (4.10) ½H(ax)= - i [ ~ , alx
F. Kirwan/ Motion of vorticeson a sphere
119
if a ~ g and by (4.9) we have ½H(ia~) = - [ f l , a ] x + i A ~ ( a ) x ,
(4.12)
where A~: g ~ g is defined by Ax( a ) . b = d/L(x)(iax)* b = ia~. ib~ = a~* bx
for all b ~ g. Thus the subspace gx + ig~ of T~X is H-invariant. If ~ ~ TxX is orthogonal to this subspace then d~(x)(~)'b=~'i~=O
for all b ~ g, so d/t(x)(,/) = 0. This implies that ½H(•) = ifl(,/) by (4.9). But the endomorphism n ~ fl(*/) of T~X vanishes on T~Z# and is nondegenerate on its orthogonal complement (T~Za).L, which contains (Tx(Stab~)Z#) ± . Therefore since TxZa is a complex subspace of T~X it suffices to show that the restriction of H to (T~(Stab ~)Z~) ± N (gx + igx) is nondegenerate. We now use the simple fact that the kernel of H is the same as the kernel of 1ill. It follows from (4.10) and (4.12) that via the obvious projection gO g--, gx + igx the restriction of ½ill to gx + igx lifts to the linear map E: g ~ g ~ g ~ g represented by the matrix
)
Ax -ad(fl)
where ad denotes the infinitesimal adjoint action of g on itself. By putting E into canonical form we find that the kernel of H restricted to gx + igx is spanned by vectors of the form a~ + ibx with
for some p > 1. Suppose that
=(:) where c ~ s t a b ~ + stabfl and d ~ s t a b f l . Since adfl and ad~ commute and are skew-adjoint we may simultaneously diagonalise them over C and deduce that ker (ad~)2(ad fl) = ker (ad ~)(ad fl) = ker (ad ~) + ker (ad fl) = stab ~ + stab fl
120
F. Kirwan/ Motion of vortices on a sphere
and ker (ad fl )2 = ker (ad r ) = stab ft.
Since (ad fl)2b = - ( a d f l ) d = 0 it follows that b ~ stabfl, and thus (ad fl)(ad ~)2a = (ad fl)(ad ~)Ax(b ) + (ad fl)(ad ~)c = (ad ~)Ax(ad fl(b)) = 0, since ad fl commutes not only with ad ~ but also with A x because fix = 0 and the metric is K-invariant. Hence a ~ stab ~ + stab ft. Therefore by induction if
for some p > 1 then a ~ stab~ + stabfl and b ~ stabfl. This shows that if I-I(a x + ibx) = 0 then a~ + ib x ~ (stabfl)x + (stab ~)~ + i(stabfl) x ~ T~(Stab ~) Z¢ and hence completes the proof of the lemma.
[]
Thus we have proved 4.13. Corollary. f is a minimally degenerate Morse function on X. Clearly f is Stab t~-invariant, and we can take the minimising manifolds ,~¢ to be Stab ~-invariant as well. Hence by [9] (10.2) f induces Stab ~-equivariant Morse inequalities relating the equivariant Betti numbers of X to those of the critical sets C¢. The proof of [9] 5.4, trivially modified, shows that f is equivariantly perfect, i.e. these equivariant Morse inequalities are in fact equalities. Thus using (4.5) and (4.6) we have 4.14. Theorem. The Stab l~-equivariant Betti numbers of C O=/~-1(~) are given by the formula
etStab~(l~-l(~))=Pt(x)et(B(Stab~))
-
E tX(a)etStab(B")(ZaN#-l(fl+ O*B~g~
where X(,8) = codim "~a is the index of f along Ca.
~))
[]
4.15. Remark. The formula of (4.14) makes sense only when the index of f along every component of Ca in X is the same. If Z a has components za, 1 Za, q, say, then we must interpret tx(a)PtStab(a'*)(Z a C~ /~-l(fl + ~)) as .....
E th(B'~)ptStab(B'~)(ZB,jNl't-l(fl+ l
C~,j = Stab I}(Z~ n tt-'(fl + ~));
~)),
F. Kirwan / Motion of vortices on a sphere
121
that is, the number i n d ( H ) of strictly negative eigenvalues of the Hessian H of f at any point x ~ CI~,j. Note that the proof of (4.11) shows that ind ( H ) = ind (i/3l (,x + i~¢.)-L) + ind (HI,x + i,x) = ind (ifl) - ind (C) + ind (Clk~re) + ind ( D ) - ind (Diker(p)), where C and D: x¢O xe+ ~O ~ are the linear maps represented by the matrices 0
-ad(fl) )
ad(fl)
0
- a d ( f l ) ),
-ad(~)
0
A~
which project to the linear maps ifl and ½H: ~¢x+ ilx--+SCx+ ix¢x via the obvious projection p: xeO ,¢--+ ,¢ + ixe. 4.16. Example. Consider again the case discussed in section 3 when X = ( S 2 ) n and K = SO(3). We can use (4.15) to compute the index k(S) of f at the point x s = (xSt. . . . . x s) where S is a subset of {1 .... , n } and (0,0,1), xS= ~ (0,0,-1),
if i ~ S , ifi*S.
Recall from (4.4) that
2 = {0} u
(k(S)-II~llISC_ {1 .....
n}}
if ~ :# 0 whereas 2 = (0} U { k ( S ) [ k ( S ) > 0 } if ~ = 0, and if 0 ~: fl ~ ~ then
Ca = Zan g-x(~+ ~)=Za--
{xSlk(S)=/~+11~11}.
To compute the index X(S) of f at x s we use the formula k ( S ) --- ind Off) - i n d (C) + ind (Clkerp) + ind ( D ) - ind (Diker(p)) of (4.15). If fl = k ( S ) - ~ is positive (when identified with a real number via the isomorphism (0, 0, x) then it is easy to check that i n d ( i f l ) = 2c(S) where c ( S ) is the cardinality of the set { j ~ S l k j > ( j ¢~ S l k j < 0}. If fl = k ( S ) - ~ is negative then ind(ifl) = 2(n - c(S)). We always have ~ = rfl for real number r, so ad(fl), ad(~) and A x all commute and thus may be simultaneously diagonalised. the eigenvalues of
-ad(~)
Ax
~ x) 0} u some Thus
F. Kirwan / Motion of vortices on a sphere
122
are the solutions h to the equations X ( X - g) + r v 2 = 0 , where (#, iv, irv) are simultaneous eigenvalues for (Ax, ad(fl), ad(~)). Hence ~ = ½(# + ~bt2- 4rv2). Since A, is always non-negative definite we have/~ > 0, so one of these values of X is always non-negative. The other is strictly negative if and only if r < 0 and v ~ 0. By assumption ~ > 0. If ~ = 0 then r = 0 and /3 > 0, while if ~ > 0 then r < 0 if and only if/3 < 0. Thus if/3 > 0 then i n d ( D ) = 0, while if/3 < 0 then ind ( D ) = dim K/Stab~3 = dim K / T = 2. A similar but easier argument shows that ind ( C ) = 2 in all cases. It is not hard to check that the index of the restriction of D to k e r ( p ) is the same as that of C in all cases, so we have proved
h(S)=
2 c ( S ) - 2,
if/3 = k ( S ) - I1~11> 0,
2(n-c(S)),
if/3=k(S)-II~ll
(4.17)
Note that if 0 :~/3 ~ ~ then Stab (/3, ~) = Stab/3 = T, so that
etStab(a'O(ZB N/x-l(/3 + ~)) = P(BT) = (1 - t 2) -1 Thus (4.4), (4.14), (4.15) and (4.17) together complete the proof of theorems 3.9 and 3.11. 4.18. Remark. We have already noted that Z~ is a Stab ~-invariant symplectic submanifold of X. Since /3, = 0 for every x ~ Z~ and /x is K-equivariant, the restriction of # to Z~ maps Z~ to the Lie algebra Stabfl of Stab/3 and is a momentum map for the action of Stabfl on Z a (cf. [9] 4.9). Therefore /x - / 3 : Za ---, Stab/3 = (Stab/3)* is also a momentum map for this action. Clearly Za (~/,-1(/3 + ~) is the inverse image of $ under this momentum map. Thus (4.14) gives us an inductive formula for the Stab ~-equivariant Betti numbers of #-1(~) (at least if bt-l(~) is not empty). Of course in the case discussed in section 3 when X= (S2)" and K = SO(3) the Poincar6 series
F. Kirwan/ Motion of vortices on a sphere
123
is easy to c o m p u t e directly, so we only need one step of the induction. I n general we m a y need several steps. T h e inductive formula given by (4.14) can be made into an explicit formula (cf. [9] 5.16) involving only the Betti n u m b e r s of X and certain fixed point sets of torus actions such as Za, together with the Betti n u m b e r s of the classifying spaces of Stab ~ and certain subgroups of Stab ~ such as Stab(fl, ~). W e have n o w shown how to c o m p u t e Ptst~b ~(#-x(~)). Finally in order to relate this to the Betti numbers of the M a r s d e n - W e i n s t e i n reductions X~ we need 4.19. Proposition. Suppose that ~ is a regular value of/x. Then the stabiliser of every x ~ # - t ( ~ ) is finite a n d the Betti numbers of the reduction X~ = / ~ - t ( ~ ) / S t a b ~ are given by
P t ( X ~ ) ~-- PtStab~(]£-l(~)) a n d thus can be c o m p u t e d using (4.14) and (4.18).
Proof. T h e first statement is an immediate consequence of the definition of a m o m e n t u m map: if a ~ xe, x ~ X and 71 ~ Tx X then d~(x)(~)
•a
=
¢Ox(~, a~)
so d # ( x ) is surjective if and only if a x 4= 0 whenever a * 0. The second statement now follows from a well-known p r o p e r t y of rational equivariant c o h o m o l o g y (cf. [9] 5.6). []
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