STABILITY FOR HOLOMORPHIC SPHERES AND MORSE THEORY
RALPH L. COHEN, JOHN D.S. JONES, AND GRAEME B. SEGAL
Abstract.In this paper we study the question of when does a closed, simp*
*ly connected, integral
symplectic manifold (X; !) have the stability property for its spaces of*
* based holomorphic spheres?
This property states that in a stable limit under certain gluing operati*
*ons, the space of based
holomorphic maps from a sphere to X, becomes homotopy equivalent to the *
*space of all continuous
maps,
lim-!Holx0(P1; X) ' 2X:
This limit will be viewed as a kind of stabilization of Holx0(P1; X). We*
* conjecture that this stabil-
ity property holds if and only if an evaluation map E : lim-!Holx0(P1; X*
*) ! X is a quasifibration.
In this paper we will prove that in the presence of this quasifibration *
*condition, then the stability
property holds if and only if the Morse theoretic flow category (defined*
* in [4]) of the symplectic
action functional on the Z - cover of the loop space, "LX, defined by th*
*e symplectic form, has a
classifying space that realizes the homotopy type of "LX. We conjecture *
*that in the presence of
this quasifibration condition, this Morse theoretic condition always hol*
*ds. We will prove this in
the case of X a homogeneous space, thereby giving an alternate proof of *
*the stability theorem for
holomorphic spheres for a projective homogeneous variety originally due *
*to Gravesen [7].
Introduction
Let (X; !; J) be a closed, connected, integral symplectic manifold of dimensi*
*on 2n with a com-
patible almost complex structure. Here ! is the symplectic 2 - form and J is th*
*e almost complex
structure. By "integral" symplectic manifold we mean that the symplectic form *
*! defines an in-
teger cohomology class, [!] 2 H2(X; Z). Recall that a map f : CP1 ! X is J - h*
*olomorphic if
df O j = J O df, where j is the almost complex structure on the tangent bundle *
*of CP1. Let x0 2 X
be a fixed based point. In this paper we consider the space of based J - holomo*
*rphic spheres,
Holx0(P1; X) = {f : CP1 ! X; such that f is J - holomorphic and f(0)}=*
* x0:
where 0 2 C C [ 1 is taken to be the basepoint of CP1. The holomorphic mappin*
*g space
Holx0(P1; X) is topologized as a subspace of the two fold loop space, 2X. The r*
*elative homotopy
type of the pair Holx0(P1; X) 2X has been studied for a variety of complex man*
*ifolds X. For
example for X = CPn, it was proved in [20] that
lim-!kHolkx0(CP1; CPn) ' 2CPn
____________
Date: May 14, 1999.
The first author was partially supported by a grant from the NSF.
The second author was partially supported by the American Institute of Mathem*
*atics.
1
2 R.L. COHEN, J.D.S. JONES, AND G.B. SEGAL
where the subscript k denotes the degree (or the homology class) of the mapping*
*, and the limit
actually refers to a homotopy colimit of spaces under the gluing of a fixed map*
* 2 Hol1x0(CP1; CPn).
Similar stability theorems have been proven for X a Grassmannian or more genera*
*l flag manifold
[8], [11], [7] X a toric variety [9], and for X a loop group, X = G, where G is*
* a compact Lie group
[7],[21]. The purpose of this paper is to begin a general investigation of what*
* basic properties of the
symplectic manifold (X; !) assure that there is an appropriate limiting process*
* so that
lim!Holx0(P1; X) ' 2X:
We will refer to this property as the stability property for the space of J - h*
*olomorphic spheres in
(X; !).
To make this question more precise, in section 1 we describe a space Holx0(P1*
*; X)+, built out of
limits of "chains" of holomorphic maps, which is an appropriate "stabilization"*
* of the holomorphic
mapping space Holx0(P1; X). To do this we will assume the following positivity *
*condition.
Definition 1.We will say that the symplectic manifold (X; !) is positive , if t*
*here is a real number
> 0 such that
for all fl 2 ss2(X).
The stabilization Holx0(P1; X)+ of Holx0(P1; X), which we will construct in t*
*he next section will
have the the following properties.
1. The inclusion of holomorphic maps into continuous maps Holx0(P1; X) ,! 2X *
*naturally
extends to a map
j : Holx0(P1; X)+ ! 2X:
2. Suppose that there are gluing operations
Holkx0(CP1; X) x Holrx0(CP1; X) ! Holk+rx0(CP1; X)
that lift (up to homotopy) the loop sum operation
2X x 2X ! 2X:
Here Holq denotes the subspace of Holx0(P1; X) consisting of classes repre*
*sented by maps
: P1 ! X with <[!]; []> = k 2 Z, where [!] 2 H2(X; Z) is the class repres*
*ented by the
symplectic form. Let 2 Holq be any fixed class with q 6= 0. Then consider*
* the map
* : Holkx0(CP1; X) ! Holk+qx0(CP1; X)
given by gluing with . Then Holx0(P1; X)+ is homotopy equivalent to the ho*
*motopy colimit
of this operation:
Holx0(P1; X)+ ' lim-!kHolkx0(CP1; X):
STABILITY FOR HOLOMORPHIC SPHERES AND MORSE THEORY *
* 3
Furthermore, if the gluing operation on Holx0(P1; X) gives it the structur*
*e of a C2- operad
space in the sense of May [14], and ss0(Holx0(P1; X)) is finitely generate*
*d as a monoid, then
Holx0(P1; X)+ is the group completion
Holx0(P1; X)+ ' B(Holx0(P1; X)):
The example to think of in this case is X a Grassmannian or more general*
* flag manifold.
Note: Since a two fold loop space is a C2 -operad space, one can take its *
*group completion,
2X+ = B(2X). But clearly 2X+ = 2X, and so in this case, when Holx0(P1; X) *
*has the
structure of a C2 - space, then the map j : Holx0(P1; X)+ ! 2X mentioned i*
*n property 1, is
simply given by applying the group completion functor to the inclusion Hol*
*x0(P1; X) ,! 2X.
We can now phrase the question posed above more precisely: What geometric pro*
*perties of a
symplectic manifold (X; !), assure that Holx0(P1; X)+ ' 2X? We conjecture an an*
*swer to this
question as follows.
Consider the evaluation map
E : Holx0(P1; X)! X
f ! f(1)
As we will see, this evaluation map naturally extends to the group completion
E : Holx0(P1; X)+ ! X:
This extension can be viewed as the composition Holx0(P1; X)+--j--!2X --E--!X*
* . Recall
that the basepoint condition is that f(0) = x0. The map E evaluates a map at th*
*e other pole, 1 2
C[1 = CP1. Recall that X is said to be rationally connected if each of the fibe*
*rs, Holx0;x(CP1; X) =
E-1(x) Holx0(P1; X) is nonempty. A stronger condition is that the map E : Holx*
*0(P1; X)+ ! X
is a quasifibration. This in particular implies that all of its fibers Holx0;x(*
*CP1; X)+ = E-1(x)
Holx0(P1; X)+ are homotopy equivalent.
Definition 2.We say that a symplectic manifold (X; !) has the "quasifibration p*
*roperty" if the
evaluation map
E : Holx0(P1; X)+ ! X
is a quasifibration.
Conjecture. A symplectic manifold has the quasifibration property if and only if
Holx0(P1; X)+ ' 2X:
4 R.L. COHEN, J.D.S. JONES, AND G.B. SEGAL
In this paper we address this conjecture using Morse theory techniques. In pa*
*rticular we use the
approach to Morse theory introduced by the authors in [4], [5], [3], which stud*
*ies the flow category,
Cf, of a function f : M ! R, where M is a smooth manifold with no boundary. Spe*
*cifically, Cf is
a topological category (where both the objects and morphisms are topologized), *
*with the objects of
Cf being the critical points of f, and the morphisms between a critical point a*
* and b, MorCf(a; b)
are given by the space of piecewise flow lines, M (a; b). If M is a closed, fin*
*ite dimensional manifold,
then in a generic setting, (i.e f : M ! R a Morse function satisfying the Morse*
* - Smale transversality
condition) the space of flow lines M(a; b) is a smooth, framed manifold of dime*
*nsion (a)-(b)-1,
where refers to the index of the critical point. M(a; b) is not compact, howev*
*er, but is, when we
adjoin all piecewise flow lines
[
M (a; b) = M(a; a1) x M(a1; a2) x . .x.M(ar; b)
where the union is taken over all sets of critical points a1; . .;.ar, such tha*
*t the above flow spaces
are nonempty. The details of this flow category, including the topology of the *
*morphism spaces, will
be reviewed in section 1. Let BCf be the classifying space (that is the geometr*
*ic realization of the
simplicial nerve) of the category Cf. The main theorem of [4] was the following.
Theorem 1. Let f : M ! R be a Morse function on a closed finite dimensional Ri*
*emannian
manifold M. Then there is a natural map OE : BCf ! M satifying the following:
1. If the Smale transversality condition is satisfied, then OE is a homeomorp*
*hism.
2. Even without the transversality condition OE is a homotopy equivalence.
In this paper we will address the above conjecture on the homotopy type of Ho*
*lx0(P1; X)+ by
studying the flow category of the symplectic area functional on the loop space *
*of a symplectic
manifold. More specifically, let (X; !) be a simply connected closed, finite di*
*mensional symplectic
manifold, and let LX denote the space of smooth loops,
LX = C1 (S1; X):
Let "LX be the Z - cover of the loop space defined by the homomorphism
(0.1) ss1(LX) = ss2(X) ! H2(X)-!---!Z
L"X = {(fl; ) 2 LX x C1 (D2; X) : |S1= fl}= ~
where the equivalence relation is given by (fl; 1) ~ (fl; 2), if the map : S2 *
*! X defined to be 1
on the upper hemisphere, and 2 on the lower hemisphere, has the property that !*
*([]) = 0 2 Z.
The symplectic area function is the map
ff : "LX! R
Z
(fl; )! *(!):
D2
STABILITY FOR HOLOMORPHIC SPHERES AND MORSE THEORY *
* 5
This is the functional that is the basis of symplectic Floer homology [6]. The*
* critical points are
pairs (x; n) 2 X x Z, where a point x 2 X is viewed as a constant loop extended*
* to a map
: (D2; S1) ! (X; x) with !([]) = n. The critical points are topologized as X x*
* Z. Moreover a
gradient flow line of ff from a critical point (x1; n1) to a critical point (x2*
*; n2) is a J - holomorphic
map fl 2 Holx1;x2(CP1; X) that represents an element in ss2(X) with !([fl]) = n*
*1 - n2 2 Z. Let
Cffbe the flow category of the symplectic area function. Notice that the above *
*theorem does not
immediately apply to the classifying space BCffbecause the loop space LX is an *
*infinite dimensional
manifold. Our main result is that this is essentially the main issue in attacki*
*ng the above conjecture.
Namely, we will prove the following theorem.
Theorem 2. Let (X; !) be a simply connected, positive, integral symplectic man*
*ifold that has the
quasifibration property. Then the map
j : Holx0(P1; X)+ ! 2X
is a homotopy equivalence if and only if the map
BCff! "LX
is a homotopy equivalence.
The paper is organized as follows. In section one we recall in more detail t*
*he construction of
the classifying space of a flow category and then define and study the stabiliz*
*ation Holx0(P1; X)+.
In section 2 we prove theorem 2. In section 3 we use this theorem to give an a*
*lternate proof of
a theorem of Gravesen [7], stating that for X a homogeneous space, its space of*
* J - holomorphic
curves has the stability property. We will continue our study of the flow categ*
*ory Cffand therefore
of the above conjecture in future work.
1. Classifying spaces and stabilized holomorphic spheres
In this section we describe the classifying space for the flow category of th*
*e symplectic action
functional, and the stabilization Holx0(P1; X)+ discussed in the introduction.
Given a smooth function f : M ! R then recall from [4] that the flow category*
* Cf has objects
the critical points of f. As mentioned in the introduction, the morphism space *
*is given by the space
of piecewise flow lines, whose definition (including its topology) we recall no*
*w.
Let OE : R ! M be a smooth curve in M. The flow equation is the differential *
*equation
(1.1) dOE_= rf(OE(t)):
dt
The curves which satisfy this equation are the flow lines of f.
6 R.L. COHEN, J.D.S. JONES, AND G.B. SEGAL
Let OE be a flow line starting at a and ending at b. Since f is strictly incr*
*easing along flow-lines
it defines a diffeomorphism of the flow-line OE with the open interval (f(a); f*
*(b)). The inverse of this
diffeomorphism gives a parametrization of the flow-line as a smooth curve
fl : (f(a); f(b)) ! M
such that
f(fl(t)) = t:
Furthermore, fl extends to a continuous function fl : [f(a); f(b)] ! M, which w*
*e will call an extended
flow-line, by setting fl(f(a)) = a and fl(f(b)) = b. We will refer to this para*
*metrisation of a flow-line,
or an extended flow-line, as the parametrization by level sets. A flow-line par*
*ametrized by level sets
satisfies the differential equation
(1.2) dfl_= _rf(OE(t))_:
dt krf(OE(t))k2
Define M (a; b) = Mf (a; b) to be the space of continuous curves fl : [f(a); *
*f(b)] ! M with the
following properties.
1. The curve fl(t) passes through at most a finite number of critical points *
*of f.
2. On the complement of the set of t where fl(t) is a critical point, the cur*
*ve fl is smooth and
satisfies equation 1.2
3. The curve fl starts at a and ends at b, that is fl(f(a)) = a and fl(f(b)) *
*= b.
This space M (a; b) is topologized as a subspace of the space C0([f(a); f(b)]; *
*M) of continuous func-
tions [f(a); f(b)] ! M with the topology of uniform convergence. A piecewise fl*
*ow-line is a curve
which satisfies these three properties and M (a; b) is the space of piecewise f*
*low-lines from a to b.
If fl is in M (a; b) then when we remove the finite set of points where fl(t) i*
*s a critical point of f
then each component of the resulting curve is a flow-line parametrized by level*
* sets. Thus a curve
in M (a; b) is a piecewise flow-line from a to b in the informal sense describe*
*d in the introduction.
Notice that there are natural pairing operations
M (a; b) x M (b;,c)! M (a; c)
fl1x fl2! fl1* fl2
where fl1 * fl2 is simply the concantenation of piecewise flows. This operatio*
*n makes Cf into a
category, where we note that the identity morphism 1 2 M (a; a) is the constant*
* flow at a.
In the case when f : M ! R is a Morse function, a specific map
OE : BCf ! M
was described, and was shown to be a homeomorphism when the Morse - Smale trans*
*versality
conditions are satisfied, and a homotopy equivalence in general. The homotopy t*
*ype of the map OE
has a rather easy description, which we now give.
STABILITY FOR HOLOMORPHIC SPHERES AND MORSE THEORY *
* 7
Given a space X Let C(X) be the category whose objects are the points of X, a*
*nd whose mor-
phisms Mor(x1; x2) are all continuous paths defined on some closed interval, th*
*at begin at x1 and
end at x2.
Mor(x1; x2) = {fi : [a; b] ! M : fi(a) = x1; fi(b) = x2; for some inte*
*rval [a;}b]:
It is a standard fact that the classifying space BC(X) is natually homotopy equ*
*ivalent to X (see
[18]).
Notice that for a smooth function f : M ! R, the flow category Cf is a subcat*
*egory of C(M).
The inclusion : Cf ,! C(M) induces on the classifying space level the map ment*
*ioned above.
Definition 3.Given a smooth function f : M ! R, define OE : BCf ! M to be the c*
*omposition
OE : BCf-B---!BC(M) ' M:
We now consider the symplectic action functional as described in the introduc*
*tion. So let (X; !)
be a closed, simply connected, integral symplectic manifold, and let LX and "LX*
* denote the loop
space and its Z - cover as defined in the introduction. Also as in the introduc*
*tion, let
ff : "LX ! R
denote the symplectic action functional, and Cffits flow category. The objects *
*in Cffare the critical
points of ff, which are given by pairs (x; n) 2 X x Z. Thus the objects of Cff*
*have a nontrivial
topology. Recall that (x; n) 2 Obj( Cff) corresponds to the constant loop at x *
*2 X, extended to
a map of a disk : (D2; S1) ! (X; x) so that !([]) = n 2 Z. As mentioned in the*
* introduction,
a flow from (x1; n1) to (x2; n2) of ff is given by a holomorphic map OE 2 Holn1*
*-n2x1;x2(P1; X), where
the subscript denotes the image under the map OE of the poles 0 and 1 in P1, an*
*d the superscript
denotes the value !([OE]) 2 Z. (See Floer's original paper [6] for details on *
*the dynamics of the
symplectic action map ff.) Thus the morphism space MorCff((x1; n1); (x2; n2)) i*
*s given by the space
of piecewise flows whose topology is as described above. We think of this spac*
*e as the space of
piecewise holomorphic maps which we denote Holn1-n2x1;x2(P1; X). An element of *
*this space can be
viewed as a chain of holomorphic maps
OE = OE1_ OE2_ . ._.OEk
where each OEi: P1 ! X is a holomorphic map satisfying the following:
1. OE1(0) = x1,
2. OEi(1) = OEi+1(0) for i = 1; . .;.k - 1
3. OEk(1) = x2
8 R.L. COHEN, J.D.S. JONES, AND G.B. SEGAL
4. The homotopy class represented by the composition
W OE1_..._OEk
OE : S2fold----!kS2------! X
has the property that !([OE]) = n1- n2 2 Z.
The space Holn1-n2x1;x2(P1; X) can be viewed as a partial compactification of*
* the holomorphic map-
ping space Holn1-n2x1;x2(P1; X), and in particular it maps to a subspace of the*
* space of stable curves as
described by Kontsevich and Manin [13]. An important difference in the topology*
* of Holx1;x2(P1; X)
and that of the Kontsevich - Manin moduli space of stable curves is that they c*
*onsider the orbits of
holomorphic maps of spheres under the action of the (holomorphic) automorphisms*
* of P1. We do
not divide out by this group of parameterizations.
Let Holnx0(P1; X) denote the union of the spaces Holnx0;y(P1; X) as y 2 X var*
*ies. It is topologized
in a natural way so that the evaluation map
(1.3) E : Holnx0(P1; X)! X
OE = OE1_ . ._.OEk! OEk(1)
is continuous. Notice that we have a continuous inclusion
Hol kx0(P1; X) ,! 2X
whose image lies in the components of 2X representing homotopy classes which ma*
*p to k 2 Z
under the homomorphism
! : ss2(X) ! Z:
Notice that there is a monoid structure on Hol*x0;x0(P1; X)
(1.4) Holn1x0;x0(P1; X) x Holn2x0;x0(P1; X) ! Holn1+n2x0;x0(P1; X)
given by concantenations of piecewise holomorphic maps:
(fl = fl1_ . ._.flr) x (OE = OE1_ . ._.OEk) ! fl1_ . ._.flr_ OE1_ . *
*._.OEk:
This also extends to give a natural action
(1.5) Holn1x0;x0(P1; X) x Holn2x0(P1; X) ! Holn1+n2x0(P1; X):
We will now use this action to define the stabilization Holx0(P1; X)+ of Holx*
*0(P1; X). As men-
tioned in the introduction, Holx0(P1; X)+ will be a certain limit of holomorphi*
*c mapping spaces.
We now make this idea precise.
Let (X; !) satisfy the positivity property 1. Choose a fixed fl 2 Holnx0;x0(*
*P1; X) with n 6= 0.
Consider the map
fl* : Holx0(P1; X) ! Holx0(P1; X)
STABILITY FOR HOLOMORPHIC SPHERES AND MORSE THEORY *
* 9
given by acting by fl as in 1.5. We define Holx0(P1; X)+ to be the homotopy co*
*limit under this
gluing map:
Definition 4.The "stabilization" Holx0(P1; X)+ of the holomorphic mapping space*
* is the homo-
topy colimit
Holx0(P1; X)+ = hoco lim-!Holx0(P1; X)
where the the homotopy colimit is taken with respect to the gluing map fl* : Ho*
*lx0(P1; X) !
Holx0(P1; X).
We observe a couple of properties of this construction. First, notice that th*
*e following diagram
homotopy commutes
Holx0(P1; X)-fl*---!Holx0(P1; X)
? ?
\?y ?y\
2X - ---!fl* 2X
where the bottom horizontal map represents the "loop sum" operation with the el*
*ement fl. Notice
furthermore that the map fl : 2X ! 2X is a homotopy equivalence, with homotopy *
*inverse given
by taking the loop sum operation with an element fl-1 2 2X representing -[fl] 2*
* ss2(X). Thus the
inclusion map Holx0(P1; X) ,! 2X extends to a map
(1.6) j : Holx0(P1; X)+ ,! 2X:
Notice furthermore that the evaluation map ( 1.3) E : Holx0(P1; X) ! X commut*
*es with the left
action by fl, in the sense that the following diagram commutes:
Holx0(P1; X)-fl*---!Holx0(P1; X)
? ?
E ?y ?yE
X - ---!= X
Thus the evaluation map extends to a map of the group completion
(1.7) E : Holx0(P1; X)+ ! X:
With these definitions (of the stabilization and the evaluation map), we may *
*now recall from
definition 2 that a closed, simply connected, integral, positive symplectic man*
*ifold (X; !) has the
quasifibration property if the evaluation map
E : Holx0(P1; X)+ ! X
is a quasifibration.
10 R.L. COHEN, J.D.S. JONES, AND G.B. SEGAL
With these definitions, the statement of our main theorem 2 is now precise. *
*This relates the
stability condition that the holomorphic mapping space Holx0(P1; X) stabilizes *
*to the continuous
mapping space 2X, to the Morse theoretic condition that the flow category of th*
*e symplectic action
realizes the homotopy type of the manifold on which it is defined, "LX. We will*
* prove the theorem
in the next section.
We end this section by making the following observation about how our stabili*
*zation construction
is related to the group completion.
Assume that X has the quasifibration property, and assume that Holx0(P1; X) h*
*as the further
property that it has the structure of a C2 - operad space, whose H - space mult*
*iplication lifts (up
to homotopy) the loop sum operation of 2X, and extends (up to homotopy) the mon*
*oid structure
of Holx0;x0(P1; X). The C2 structure assures that the monoid ss0(Holx0(P1; X))*
* is commutative,
and that the topological monoid Holx0(P1; X) is homotopy commutative. Assume f*
*urthermore
that ss0(Holx0(P1; X)) is finitely generated as a monoid. Let {fl1; . .;.flk} b*
*e a set of elements in
Holx0(P1; X) that generate ss0(Holx0(P1; X)). Then in our definition of Holx0(P*
*1; X)+ we can take
our "gluing map" fl = fl1+. .+.flk: Then by the group completion theorem [15] w*
*e have a homology
equivalence
(1.8) Holx0(P1; X)+ ' B(Holx0(P1; X)):
This structure (i.e (X; !) a positive, integral symplectic manifold, with Holx0*
*(P1; X) a C2 - operad
space, with finitely generated ss0) exists, for example, when X is a coadjoint *
*orbit of a compact
Lie group on its Lie algebra, or when X is a loop group X = G, where G is a sim*
*ply connected
compact Lie group (see [1], [23]).
2.Proof of theorem 2
In this section we prove the main theorem 2.
Proof.Throughout we will assume that X has the quasifibration property. Let Spa*
*ces denote the
category of based topological spaces and basepoint preserving continuous maps. *
*Consider the functor
H : Cff! Spaces
which on objects is defined by H((x; g)) = Holx0;x(P1; X)+, by which we mean th*
*e fiber at x 2 X
of the evaluation map E : Holx0(P1; X)+ ! X. On the level of morphisms, if fl 2*
* Holx1;x2(P1; X),
then H(fl) is the gluing operation (on the right)
(2.1) H(fl) : Holx0;x1(P1;-X)+*fl---!Holx0;x2(P1; X)+:
Notice that by the quasifibration property we have the following:
STABILITY FOR HOLOMORPHIC SPHERES AND MORSE THEORY *
* 11
Lemma 3. For every morphism fl 2 Holx1;x2(P1; X), the induced map
H(fl) : Holx0;x1(P1;-X)+*fl---!Holx0;x2(P1; X)+:
is a homotopy equivalence.
This action of the morphisms of Cffon the functor H, H(x1; g1) x Mor((x1; g1)*
*; (x2; g2)) !
H(x2; g2), we write as
H xob(Cff)Mor(Cff) ! Mor(Cff):
We now consider the following "simplicial Borel construction", ECff(H), whose*
* n - simplices are
given by
ECff(H)n = H xob(Cff)Mor(Cff) xob(Cff)Mor(Cff) xob(Cff).x.o.b(Cff)Mor(*
*Cff);
n - copies of Mor(Cff). The notation Mor(Cff) xob(Cff)Mor(Cff) refers to taking*
* products of compos-
able morphisms. The face maps are defined as usual by composition of morphisms *
*and by the action
of the morphisms on the functor H. The degeneracy maps are defined by insertin*
*g the identity
morphism in the various slots. Notice that the projection maps on the level of *
*n - simplices,
pn : ECff(H)n! (BCff)n
H xob(Cff)Mor(Cff) xob(Cff).x.o.b(Cff)Mor(Cff)! Mor(Cff) xob(Cff).x.o.b(Cf*
*f)Mor(Cff)
fit together to give a map of simplicial spaces
p : ECff(H) ! BCff
Our main technical result needed to prove theorem 2 is the following.
Theorem 4. The induced map on the level of geometric realizations,
p : kECff(H)k ! kBCffk
is a quasifibration with fiber H(x0) = Holx0;x0(P1; X)+. Furthermore the space *
*kECff(H)k is con-
tractible.
The following is an immediate consequence.
Corollary 5.There is a natural homotopy equivalence
j : Holx0;x0(P1; X)+'----!BCff:
Proof.( theorem 4). The first part of the theorem follows from lemma 3 and the *
*following lemma,
which was proven in [19].
12 R.L. COHEN, J.D.S. JONES, AND G.B. SEGAL
Lemma 6. If p : E ! B is a map of simplicial spaces such that Ek ! Bk is a qua*
*sifibration for each
k 0, and for each simplicial operation : [k] ! [l] and each b 2 Bl the map p-*
*1(b) ! p-1(*(b))
is a homotopy equivalence, then the map of realizations kEk ! kBk is a quasifib*
*ration.
To prove the second part of the theorem we observe that the simplicial space *
*ECff(H) is the
nerve (classifying space) of the topological category Cff(H) whose objects are *
*elements of the space
Holx0(P1; X)+, and if fl1 2 Holx0;x1(P1; X)+ and fl2 2 Holx0;x2(P1; X)+ are obj*
*ects in ECff(H), then
a morphism OE : fl1 ! fl2 is an element of MorCff((x1; n1); (x2; n2)) for some *
*n1; n2 2 Z, such that
under the gluing operation 2.1
fl1* OE = fl2:
Furthermore, notice that the category Cff(H) has an initial object: namely the *
*constant holomor-
phic map fflx0 2 Holx0;x0(P1; X)+. Thus the realization of its classifying spa*
*ce (~= kECff(H)k) is
contractible. The theorem, and hence the corollary follow. *
* |___|
We remark that 4 is a kind of "group completion theorem" of the sort original*
*ly proved in [15].
Generalizations of the sort proved here (done in the category of bisimplicial s*
*ets) were done in [16],
[10], and [22]. This theorem will be useful in our proof of 2, which we now com*
*plete.
Recall from section 1 the definition of the map
OE : BCff! "LX
via the category C("LX). This category has objects points of "LX, and morphisms*
* are paths in "LX.
Since an element of "LX is given by a pair (fl; ), where fl 2 LX, and : D2 ! X*
* is a homotopy
class of an extension of fl, then a morphism in C("LX) from (fl0; 0) to (fl1;*
* 1), can be viewed as
a map of the cylinder
: S1 x [0; 1] ! X
with 0 = fl0 and 1 = fl1. In particular, if (fflx0; 0) and (fflx1; 1) are the*
* objects in C("LX) corre-
sponding to the constant loops at x0 and x1 and extensions 0; 1 : S2 ! X with !*
*([i]) = ni, then
a morphism in C("LX) between them are given by elements of the mapping space Ma*
*pn0-n1x0;x1(S2; X).
Recall that the map OE : BCff! "LX was given by the composition
OE : BCff-j---!BC("LX) ' "LX
where j : Cff,! C("LX) is the inclusion of categories. On morphism spaces the f*
*unctor j is given by
including spaces of holomorphic maps into spaces of continuous maps. Now by per*
*forming the same
argument used to prove 4, but replacing the category Cffby C("LX), we see that *
*this implies that we
have the following homotopy commutative diagram:
STABILITY FOR HOLOMORPHIC SPHERES AND MORSE THEORY *
* 13
BCff --OE--! "LX
? ?
(2.2) '?y ?y'
Holx0;x0(P1; X)+----!Mapx0;x0(S2; X):
j
Now consider the evaluation map E : Holx0(P1; X) ! X. By assumption, this map*
* is a quasifi-
bration. In fact this means that the inclusion j of holomorphic maps into conti*
*nuous maps induces
a map of quasifibrations
Holx0;x0(P1; X)+-j---!Mapx0;x0(S2; X)
? ?
\?y ?y\
Holx0(P1; X)+ --j--! 2X
? ?
E?y ?yE
X ----!= X:
Thus j : Holx0(P1; X)+ ! 2X is a homotopy equivalence if and only if the indu*
*ced map of
fibers j : Holx0;x0(P1; X)+ ! Mapx0;x0(S2; X) is a homotopy equivalence. But by*
* 2.2 this map is
a homotopy equivalence if and only if OE : BCff! "LX is a homotopy equivalence.*
* This proves the
statement of theorem 2. *
* |___|
3. Homogeneous Spaces
In this section we use theorem 2 to give an alternative proof to the followin*
*g stability theorem of
Gravesen [7].
Theorem 7. Let G be a complex linear algebraic group, and let P < G be a parab*
*olic subgroup.
The homogeneous space G=P has the structure of a smooth projective variety. The*
*n G=P has the
stability property:
Holx0(P1; G=P )+ ' 2G=P:
Remark.
1. Gravesen stated his result in terms of the colimit under a gluing operatio*
*n on Holx0(P1; G=P ):
lim-!Holx0(P1; G=P ) ' 2G=P:
By the construction of Holx0(P1; G=P )+ in section 1, it is clear that thi*
*s limit is the same as
our stabilization Holx0(P1; G=P )+.
14 R.L. COHEN, J.D.S. JONES, AND G.B. SEGAL
2. In [1] Boyer, Hurtubise, Mann, and Milgram used Gravesen's theorem to prov*
*e a stronger sta-
bility theorem. Namely given n 2 Z, they showed that there is an explicit *
*range of dimensions
that increases over the limiting process, in which the inclusion
j* : ssq(Holnx0(P1; G=P )) ! ssq(2G=P )
is an isomorphism.
By theorem 2, to prove 7 it suffices to prove the following results:
Proposition 8.G=P has the quasifibration property.
Proposition 9.Let Cff(G=P ) be the flow category of the symplectic action funct*
*ional on "L(G=P ).
Then the map
OE : BCff(G=P ) ! "L(G=P )
is a homotopy equivalence.
We will prove proposition 9 first.
Proof.For P < G a parabolic subgroup, G=P is a compact, smooth projective varie*
*ty. Let e :
G=P ,! PN be a projective embedding. The symplectic form on G=P is the pull b*
*ack under
this embedding of the canonical symplectic form on PN . Now ss2(PN ) = Z, and t*
*he symplectic form
! : ss2(PN ) ! Z is an isomorphism. Thus "LPN is the universal cover of the loo*
*p space LPN . "L(G=P )
is the pull back of the cover "LPN under the induced embedding of loop spaces, *
*e : L(G=P ) ,! LPN .
Notice that the symplectic action functional on projective space,
ff : "LPN ! R
yields the symplectic action on G=P via the composition
(3.1) ff : "L(G=P-)e---!"L(PN-)ff---!R:
This induces a functor between the corresponding flow categories:
(3.2) Cff(G=P )-e---!Cff(PN ):
In order to study the flow category Cff(G=P ) we will use this functor togeth*
*er with a study of
Cff(PN ). This category was studied in detail in [5]. We recall those results n*
*ow.
Let W = C[z; z-1] be the vector space of Laurent polynomials, topologized as *
*a space of maps
Cx ! C, where Cx = C - {0}. The linear flow zk ! ektzk defines a flow on the i*
*nfinite projective
space P(CN+1 W ). This in fact is a gradient flow. The stationary points of a*
*re PN x Z, where
PN x {k} is the subspace P(CN+1 zk) P(CN+1 W ). If Wmnis the subspace of W s*
*panned
by the zj's with m j n, then the space of points which lie on piecewise flow *
*lines of which
STABILITY FOR HOLOMORPHIC SPHERES AND MORSE THEORY *
* 15
go from level n to level m is P(CN+1 Wmn) = P(N+1)(n-m)-1, which is compact. I*
*n fact it was
stressed in [5] that the flow category C is a compactification of Cff(PN ) in *
*the following sense. Let
UN be the open dense subset of P(CN+1 W ) consisting of (N + 1) - tuples of La*
*urent polynomials
(p0; . .;.pN ) 2 CN+1 W with no common roots in Cx. (By common roots we mean r*
*oots common
to all the polynomials {p0; . .;.pN }.) UN is invariant under the flow , and it*
* was seen easily that
the corresponding flow category is isomorphic to the category Cff(PN ). Thus we*
* have the inclusion
of flow categories, Cff(PN ) C . Moreover, since the flow on P(CN+1 W ) is t*
*he limit of flows
on the finite dimensional compact manifolds P(CN+1 Wmn) = P(N+1)(n-m)-1, then *
*we know that
BC ~=P(CN+1 W ). Moreover the realization of the inclusion functor BCff(PN ) *
* BC gives the
inclusion of the open dense subset UN P(CN+1 W ).
In particular if Cn;m is the full subcategory of C whose objects are PN x {m*
*; m + 1; . .;.n}, then
by the results of [4] the map
OE : BCn;m ! P(CN+1 x Wmn)
is a homotopy equivalence, where the inverse image of each point is a simplex. *
*Taking the limit over
n and m, we have that
(3.3) OE : BC ! P(CN+1 x W )
is a proper map and a homotopy equivalence, where the inverse image of each poi*
*nt is a simplex.
The pull back of OE to UN P(CN+1 x W ) is the map BCff! UN , which is therefor*
*e a homotopy
equivalence.
Now it was seen in [5] that the map OE : BCff(PN ) ! "L(PN ) can be realized *
*in terms of the space
UN , by observing that an (N + 1) - tuple of Laurent polynomials
(p0; . .;.pN ) 2 UN P(CN+1 W )
determines a map
p : Cx ! PN :
Since this map is algebraic it extends to a holomorphic map p : C [ 1 ! PN . By*
* restricting this
holomorphic map to the equator (i.e the unit circle S1 C) we get an element of*
* LPN . Using the
given extension of this map to the unit disk, this actually defines an element *
*of "LPN . This defines
an embedding UN ,! "LPN , which makes the following diagram homotopy commute:
BCff(PN )-OE---!"LPN
? ?
(3.4) '?y ?y=
UN ,! "LPN :
This describes the homotopy type of the classifying space BCff(PN ) as realizin*
*g the subspace of
"L(PN ) consisting of those loops whose projective coordinates have finite Four*
*ier expansions (i.e
are Laurent polynomials). We see that by restricting to Cff(G=P ), we get a sim*
*ilar description for
BCff(G=P ) as follows.
16 R.L. COHEN, J.D.S. JONES, AND G.B. SEGAL
The projective embedding e : G=P ,! PN is defined by a homogeneous ideal I(G=*
*P ) C[x0; . .;.xn].
Then consider the subspace U(G=P ) UN P(CN+1 W ) defined by
(3.5) U(G=P ) = {(p0; . .;.pN ) 2 UN : f(p0; . .;.pN ) = 0 for all f 2 I(*
*G=P}):
Notice that U(G=P ) UN is a - flow invariant subspace. Its stationary point*
*s correspond to
G=P x Z Pn x Z, and the points lying on flows going from level n to level m ar*
*e given by (N + 1)
- tuples, (p0; . .;.pN ) P(CN+1 Wmn) with f(p0; . .;.pn) = 0 for all f 2 I(G=*
*P ). This space
precisely parameterizes the holomorphic maps fl : P1 ! G=P so that the composit*
*ion P1 ! G=P ,!
PN has degree n - m. Thus the flow category of U(G=P ) is isomorphic to Cff(G=P*
* ). In particular
the pull back of the map OE : BC ! P(CN+1 x W ) to the subspace U(G=P ) UN P*
*(CN+1 x W )
is precisely BCff(G=P ). Since OE is a proper map and a homotopy equivalence wi*
*th the inverse image
of each point a simplex (3.3), then the pull back BCff(G=P ) ! U(G=P ) is a hom*
*otopy equivalence.
Note also that the image of U(G=P ) UN ,! "LPN lies in "L(G=P ) and consists p*
*recisely of those
loops whose homogeneous coordinates have finite Laurent expansion. So we have *
*the following
commutative diagram:
BCff(G=P )---e-! BCff(PN )
? ?
'?y ?y'
(3.6) U(G=P ) ,! UN
? ?
\?y ?y\
"L(G=P )----! "L(PN )
e
Moreover the vertical compositions in this diagram are homotopic to the maps OE*
* : BCff(G=P ) !
"L(G=P ) and OE : BCff(PN ) ! "L(PN ). Thus to prove that OE : BCff(G=P ) ! "L(*
*G=P ) is a homotopy
equivalence it suffices to prove the following.
Lemma 10. The inclusion U(G=P ) ,! "L(G=P ) is a homotopy equivalence.
Remark. As observed above, this inclusion can be viewed as the inclusion of the*
* space of polynomial
loops (i.e those loops which, in homogeneous coordinates have finite Fourier ex*
*pansion) into the space
of all loops.
Proof.Since G is a linear algebraic group, it is a subgroup of GL(N; C) for som*
*e N. This N can be
taken to be the dimension of the projective embedding e : G=P ,! PN . So in par*
*ticular a loop in
G can be viewed as a loop in the affine space of matrices. Let LG be the infini*
*te group of smooth
loops in G, and LpolG the subgroup of polynomial loops; those loops in G which,*
* together with
STABILITY FOR HOLOMORPHIC SPHERES AND MORSE THEORY *
* 17
their inverses have finite Fourier expansion. That is, they are given by finite*
* Laurent polynomials.
Notice that L(GL(N; C)) acts transitively on "L(PN ), and the action restricts *
*to a transitive action
of LG on "L(G=P ). Furthermore the isotropy subgroup of a constant loop is clea*
*rly a union of path
components of the loop group of P , which we denote by L(P )0. This subgroup o*
*f L(P ) has the
property that the quotient group is infinite cyclic and that the projection map*
* p : L(G)=L(P )0 !
L(G)=L(P ) = L(G=P ) is the infinite cyclic cover p : "L(G=P ) ! L(G=P ):
Now the polynomial loop group Lpol(GL(N; C)) acts transitively on the space o*
*f polynomial
loops UN L(PN ). It restricts to give a transitive action of Lpol(G) on U(G=P *
*). In this case the
isotropy subgroup of a constant loop is th union of path components Lpol(P )0 o*
*f Lpol(P ) corre-
sponding to the subgroup L(P )0 of L(P ). We have therefore shown that there is*
* a homeomorphism
LpolG=Lpol(P )0 ~=U(G=P ) making the following diagram commute:
~=
LpolG=Lpol(P )0----!U(G=P )
? ?
(3.7) \?y ?y\
L(G)=L(P )0 ----!~ "L(G=P ):
=
Now it was proved in chapter 8 of [17] that the inclusion of the polynomial loo*
*p group into the smooth
loop group Lpol(G) ,! L(G) is a homotopy equivalence. (This proof also uses Mor*
*se theory!) Since
this also holds for Lpol(P ) ,! L(P ) it also holds for the corresponding union*
* of connected compo-
nents Lpol(P )0-'---!L(P )0. This implies that the inclusion LpolG=Lpol(P )0 ,!*
* L(G)=L(P )0 is a
homotopy equivalence. By by this diagram this implies that the inclusion U(G=P *
*) ,! "L(G=P ) is a
homotopy equivalence. This proves the lemma.
*
*|___|
Now as remarked earlier, the proof of the lemma completes the proof of proposit*
*ion 9. |___|
We now prove proposition 8. As seen earlier, this is the last step in the pro*
*of of theorem 7.
Proof.We need to show that for X = G=P , the evaluation map
E : Holx0(P1; X)+ ! X
is a quasifibration. Let x1 2 X. It is well known that the homogeneous space X *
*= G=P is rationally
connected (see for example [12]) so Holx0;x1(P1; X) is nonempty. Let 2 Holx0;x*
*1(P1; X) be any
element. Gluing with induces a map * : Holx0;x0(P1; X)+ ! Holx0;x1(P1; X)+. *
*It suffices to
show that
* : Holx0;x0(P1; X)+ ! Holx0;x1(P1; X)+
is a homotopy equivalence. Now since X = G=P is rationally connected, Holx1;*
*x0(P1; X) is
nonempty. Let ff 2 Holx1;x0(P1; X). We will prove that gluing with the products,
(ff * )* : Holx0;x0(P1; X)+ ! Holx0;x0(P1; X)+
18 R.L. COHEN, J.D.S. JONES, AND G.B. SEGAL
and
( * ff)* : Holx0;x1(P1; X)+ ! Holx0;x1(P1; X)+
are homotopy equivalences. Now for X = G=P , the holomorphic mapping space Holx*
*0(P1; X) is a
C2 - operad space. The structure is studied, for example, in [1]. This in parti*
*cular implies that the
monoid structure in Holx0;x0(P1; X) is homotopy commutative. Furthermore ss0(Ho*
*lx0(P1; X)) is
finitely generated [1], [7]. So by 1.8 this implies that Holx0(P1; X)+ is group*
* complete, and so in par-
ticular ss0(Holx0(P1; X)+) is a group. But since X is rationally connected, ss0*
*(Holx0;x0(P1; X)+) ~=
ss0(Holx0(P1; X)+). Thus ss0(Holx0;x0(P1; X)+) is a group. This means that th*
*e class [ff * ] 2
ss0(Holx0;x0(P1; X)+) has an inverse, and so the element ff * 2 Holx0;x0(P1; X*
*)+ has a homotopy
inverse. This means that (ff*)* : Holx0;x0(P1; X)+ ! Holx0;x0(P1; X)+ is a homo*
*topy equivalence.
Now X = G=P is homogeneous, so x1 = gx0 for some g 2 G. Multiplication by g i*
*s a holomorphic
map from G=P to itself, and so the class * ff 2 Holx1;x1(P1; X) determines an *
*element g . ( *
ff) 2 Holx0;x0(P1; X). As argued above, this class has a homotopy inverse whic*
*h we call fi 2
Holx0;x0(P1; X)+. Notice then that g-1 . fi 2 Holx1;x1(P1; X)+ defines a gluing*
* map
(g-1 . fi)* : Holx0;x1(P1; X)+ ! Holx0;x1(P1; X)+
which is a homotopy inverse of ( *ff)*. Thus ( *ff)* is a homotopy equivalence *
*as well. This proves
the proposition and thereby completes the proof of theorem 7. *
* |___|
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Dept. of Mathematics, Stanford University, Stanford, California 94305
E-mail address, Cohen: ralph@math.stanford.edu
Department of Mathematics, University of Warwick, Coventry, England
E-mail address, Jones: jdsj@maths.warwick.ac.uk
Dept. of Pure Mathematics and Mathematical Statistics, Cambridge University, *
*Cambridge, England
E-mail address, Segal: G.B.Segal@dpmms.cam.ac.uk