Dec 15, 1995 - following is a standard result in classical Morse theory. See 0] for example. ...... spaces Stanford University preprint, 1991. 3] R.L. Cohen, J.D.S. ...
1
1
Graph Moduli Spaces and Cohomology Operations Martin Betz Ralph L. Cohen December 15, 1995 Abstract
In this paper we de ne a moduli space of \Graph ows" in a manifold, and use them to de ne analogues of Donaldson invariants. These take values in tensor products of the cohomology and homology of the manifold and can be interpreted as generalized cohomology operations. We show how to construct classical invariants such as the Stiefel - Whitney classes and the Steenrod operations in this way. We also give homotopy theoretic descriptions of these invariants which will allow the de nition of higher order Donaldson type invariants.
The polynomial invariants de ned by Donaldson [0] have had a dramatic impact on four dimensional di erential topology in recent years. The simplest, zero degree invariant q0(X ), of a closed, simply connected, smooth Riemannian four manifold X , is an integer which is given by counting (with sign) the components of M0(X ), the zero dimensional moduli space of antiself dual connections on an SU (2) bundle over X . (Recall that the dimension of Mk (X ), the moduli space of ASD connections on the principal SU (2) bundle of Chern class k 2 H 4(X ) = Z, is 8k ? 3(1 + b+(X )) where b+ (X ) is the rank of the maximal positive de nite subspace of the intersection form.) For manifolds with boundary, the analogous invariants take values in the Floer (co)homology of the boundary. For example, if X is a four manifold with boundary a homology 3 - sphere, say @X = Y , then the zero degree invariant q0(X ) 2 HF(Y ). Roughly, this invariant is de ned as follows.
The second author was partially supported by a grant from the NSF
2
Given a connection 2 M0(X ) with nite Yang - Mills energy, then it approaches a at connection, say ( ), on the trivial SU(2) bundle over Y . After suitable perturbations of the metric this limiting connection can be viewed as an element of the chain complex CF(Y ), of Floer's \instanton homology" of Y . (A sign is assigned to the at connection coming from orientation considerations.) One can then de ne the resulting Donaldson invariant by X q0(X ) = ( ) 2 CF(Y ):
2M0 (X )
It is proved in [0] that this class is a cycle in the Floer chain complex and determines an element q0(X ) 2 HF(Y ) which is independent of the choices of perturbations and metrics, and is an invariant of the smooth structure on X . (Actually whether q0(X ) lies in Floer homology or Floer cohomology depends on choice of normal vector eld of Y .) Now if X is an oriented four manifold with several boundary components, say @X = Y1 t t Yk where the Yi's are homology spheres, then the Donaldson invariant takes values in the tensor product
q0(X ) 2 HF (Y1) HF (Yj ) HF(Yj+1 ) HF(Yk ) where Y1; Yj are the boundary components whose normal bundles are oriented with an inward pointing vector eld, and Yj+1 Yk are the boundary components oriented with outward pointing vector elds. When we take eld coecients and use the identity
HF HF = Hom(HF ; HF )
we can think of these Donaldson invariants as operations in Floer cohomology. Now similar invariants exist in the setting of pseudo-holomorphic curves in a symplectic manifold. These were rst de ned in [0] [0]. More speci cally, let (M 2n; !), be a closed, simply connected symplectic manifold. Let be a 3
Riemann surface with k - boundary components. Then let M(; M ) be the moduli space of pseudo - holomorphic maps
: ?! M which when restricted to a boundary component circle is a constant loop in M . Let M0 denote the zero dimensional component of this space. After suitable perturbations, an element 2 M0 determines an element of the tensor product of Floer chain complexes,
( ) 2 CF(M ) k given by the restriction of to the boundary components. Here CF is the Floer chain complex of the symplectic action functional on the loop space L(M ). Again, the Donaldson invariant is de ned by counting with sign
q0(; M ) =
X
2M0
( ) 2 CF(M ) k :
which is shown to be a cycle in [0], and represents an element
q0(; M ) 2 HF(M ) k which is an invariant of the symplectic structure on M . Again, this invariant may be interpreted as an operation on the Floer cohomology. This point of view was introduced by Witten in [0] in which he used this invariant to construct cup product structures on the Floer cohomology of the symplectic action on the loop space LC P n . Our goal is to understand in more depth the relationship between Donaldson invariants, and more generally, the homotopy type of thes types of moduli spaces \with ends", and the Floer homology (and more generally the \Floer homotopy type" in the sense of [0]) of the ends. In order to understand the basic structure of this relationship we will examine an example of a \moduli space with ends" where the resulting \Floer cohomology" of the ends is usual cohomology. This is a moduli space of graphs in a closed manifold, which was de ned and studied in [0]. We will describe some of the results of [0] here and expand upon them. In particular we will show how the classical operations in the cohomology of a manifold, 4
including characteristic classes and the Steenrod operations can be obtained as Donaldson invariants in this theory. We will also describe a homotopy theoretic approach to these Donaldson invariants which will allow us to view secondary (and higher order) Steenrod operations as invariants de ned by these moduli spaces. The goal is to use this approach to de ne secondary and higher order Donaldson invariants for four manifolds and symplectic manifolds when certain primary Donaldson invariants vanish. This will be studied in a future paper. This paper is organized as follows. In section one we will de ne the moduli space of graphs, the resulting Donaldson invariants, and prove a basic gluing theorem. Some examples will also be discussed. In section two we show how to produce equivariant analogues of these invariants using the symmetry groups of these graphs and show how the Steenrod operations are examples. These two sections summarize some of the results of the rst author in his Ph.D thesis [0]. In section three we describe the beginning of a joint project in which we apply the categorical and homotopy theoretic viewpoint of Morse theory and Floer theory developed by the second author, J. Jones, and G. Segal [0], [0] to study these invariants and to de ne higher order invariants. The second author would like to thank the organizers of the Gokova conference for the opportunity to participate in an exciting conference in a beautiful part of the world.
5
1 The Moduli Space of Graph Flows
Let M be a closed, compact, smooth Riemannian manifold of dimension d, and let Let ? be an oriented, nite, possibly non-compact, graph with m edges parametrized by [0; 1], (?1; 0], and [0; 1). We call these edges \internal", incoming", and \outgoing" respectively. Let these edges be indexed fE1; :::; Emg such that the rst n are noncompact, and the rest are internal. Among the n noncompact edges the rst n1 are assumed to be incoming, the next n2 = n ? n1 are are assumed to be outgoing. In this section we de ne the moduli space M(?; M ) of \graph ows", and use these to de ne the analogues of Donaldson invariants mentioned in the introduction. We show that these invariants can be viewed as operations in H (M ), and compute some examples. We begin by de ning the notion of an M -structure for our graph ?. The space of all such M -structures will play a signi cant roll in our constructions.
De nition 1 Fix an oriented, parameterized graph ? and a closed Rieman-
nian manifold M as above. An M -structure on ? consists of the following: 1. A real number `i associated to each internal edge of Ei of ?. We think of `i as the length of Ei, even though we allow `i 0. 2. A function fi 2 C 1 (M ) associated to each edge Ei of ?. We assume the fi 's are distinct.
The space of all M -structures will be denoted S (?; M ). Notice that there is a homeomorphism
S (?; M ) = Rm?n Fm(C 1(M ))
where Fm(X ) X m is the con guration space of m distinct points in X . For xed choice of such a structure , we are now ready to de ne the moduli space M (?; M ) of \? - ows in M ". let : ? ! M be a continuous map, smooth on the edges. For each internal edge Ei let i : [0; 1] ! M be the restriction of to Ei composed with the parameterization of Ei by [0; 1] given as part of the data of ?. For the incoming and outgoing edges we de ne i : (?1; 0] ! M or i : [0; 1) ! M similarly. 6
De nition 2 lies in M (?; M ) if and only if for each edge Ei it satis es
the di erential equation
d i=dt + `i rfi = 0: For the noncompact edges (i.e the incoming and outgoing edges) in this equation `i is assumed to be 1. Here rfi is the gradient vector eld. M (?; M ) is topologized as a subspace of C 0(?; M ).
We let M(?; M ) be the union of the spaces M (?; M ) where the structures vary in S (?; M ). M(?; M ) is topologized so that natural the projection map : M(?; M ) ! S (?; M ) is continuous. Given P S (?; M ), let MP (?; M ) = ?1(P ). These spaces will be important in general, but in this chapter we restrict ourselves to studying M (?; M ), the moduli space associated to a single structure. We now describe some basic properties of these moduli spaces. These properties are analogues of properties of moduli spaces of anti-self dual connections and of pseudo-holomorphic curves, and have similar (in fact easier) proofs. See [0] for details. Again, x a structure 2 S (?; M ). This de nes a vector of labelling functions of the edges. Let f = (f1; :::fn) be the n - tuple of functions labelling the noncompact edges. Observe that every 2 M (?; M ) has the property that its restriction to each noncompact edge i is a gradient ow line, so it therefore converges to a critical point, say ai, of the function fi. Thus can be associated to an n - tuple ~a = (a1; ; an) where ai is a critical point of fi. For a xed n - tuple ~a, let
M (?; M ; ~a) M (?; M ) be the subspace of those 2 M (?; M ) which converge on the ith edge to
the critical point ai.
7
Theorem 1 For a generic choice of structure 2 S (?; M ), the moduli spaces M (?; M ; ~a) are manifolds for every n- tuple of critical points ~a. The dimension of M (?; M ; ~a) is given by the formula 1 [index(a )] ? Pn2 [index(a dim(M (?; M ; ~a)) = Pni=1 i n1 +i )] ? d(n1 ? 1) i=1 ? d dim(H1(?; R)) where, as above, n1 and n2 are the number of incoming and outgoing edges of ? respectively. Furthermore an orientation on the manifold M induces orientations on the moduli spaces M (?; M ; ~a).
The generic condition on the structure in this theorem is that the labelling functions fi are Morse, and satisfy the \Morse - Smale transversality properties". That is, the stable and unstable manifolds of the critical points all intersect transversally. Fix a structure satisfying this generic property. We will now construct a natural compacti cation of the space M (?; M ; ~a). To do this we rst recall the natural compacti cation of the space of gradient ow lines of a Morse function converging to two xed critical points. We will refer to the space of ow-lines from critical point ai to critical point bi by M (ai; bi). The following is a standard result in classical Morse theory. See [0] for example.
Lemma 2 Let M(a; b) denote the space of \piecewise ow lines" connecting
connecting critical points a and b. That is
M(a; b) =
[
a=a0 >a1 >:::>aj =b
M(a; a1) ::: M(aj?1; b);
where the union is taken over decreasing nite sequences of critical points. (The partial ordering is de ned by i M( ; ) is nonempty.) Then M(a; b) is compact and contains M(a; b) as an open dense subspace.
There is a similar compacti cation for the moduli spaces of ? - ows. Namely, let [ M (?; M ; ~a) = M (?; M ; ~b) M (b1; a1) M (an ; bn): ~b
8
Whether we use Ms(bi; ai) or Ms(ai; bi) in the above union depends on whether the ith edge is incoming or outgoing. M (?; M ; ~a) consists of ? ows that are allowed to be piecewise ows on the noncompact edges. We refer to these as \piecewise ? - ows". There is an obvious way to topologize M (?; M ; ~a). The proof of the following is simply an adaptation of the proof of the above lemma, and is carried out in [0].
Theorem 3 The space M (?; M ; ~a) is compact and contains M (?; M ; ~a) as an open dense subspace.
This result can be improved in such a way so as to identify the ends of the moduli space M (?; M ; ~a). To do this we set up the following notation. For n - tuples of critical points ~a and ~b associated to the structure , consider the oriented spaces of ow lines
Mi = Mf (bi; ai) for incoming Ei Mi = Mf (ai; bi) for outgoing Ei: i
i
Theorem 4 There exist \gluing" maps Y Mi [0; 1) ! M (?; M ; ~b); ~a;~b : M (?; M ; ~a) ai 6=bi
that are orientation preserving homeomorphisms onto disjoint images. Moreover the complement of the images, [
M (?; M ; ~b) ? ~a;~b ~a
is compact.
In this paper we will be primarily concerned with the moduli spaces of dimension zero and one, M0 (?; M ; ~a) and M1 (?; M ; ~a). These theorems tell us that M0 (?; M ; ~a) = M0 (?; M ; ~a) is a nite set of points with signs (orientation). Moreover if an end of one of these isolated ? - ows glues to 9
an isolated ow line, then the pair forms one end of a compact interval of ? - ows. The other end of this interval is modelled by another such pair. This information will allow us to de ne a Donaldson - type invariant for these moduli spaces, which we now proceed to do. Fix a generic structure 2 S (?; M ) as above. Given our Morse-Smale functions fi, let C(M; fi ) be the associated Morse-Smale chain complex generated by the critical points, and let C (M; fi ) be the dual cochain complex. We de ne a class q(?; M ) to be an element of the complex O
1in1
O
C (M; fi )
n1 +1in
C(M; fi )
in the following manner. Consider those n - tuples of critical points ~a such that dim(M (?; M ; ~a)) = 0. These spaces contain a nite number of oriented points which can be counted with sign (if M is oriented - otherwise this is well de ned mod 2, and we take coecients to be Z2).
De nition 3 q(?; M ) =
X
#M (?; M ; ~a)[~a] 2
O 1in1
C (M; fi )
O
n1 +1in
C(M; fi ):
Using the gluing theorem above and the de nition of the boundary and coboundary operators in the Morse-Smale complex, one can show the following (see [0]):
Lemma 5
dq = 0:
We shall therefore view q(?; M ) as an element of the associated homology,
q(?; M ) 2 H (M ) n1 H (M ) n2 : The following says that q(?; M ) is indeed an invariant of M . 10
Theorem 6 The homology class q(?; M ) does not depend on the choice of structure 2 S (?; M ). Sketch of Proof: Since the space of generic structures inside S (?; M )
is connected we can nd curves connecting any two generic structures. The induced paths give chain homotopy equivalences that preserve the q(?; M )'s
We now describe four basic examples of these invariants. Example 1. ? = In this case M (?; M ; ~a) has dimension zero if and only if ~a = (a) is a maximum. Thus q(?; M ) 2 Hd(M ), and it can easily be seen to be the fundamental class. (Coecients should be taken in Z2 if M is not orientable). Example 2. ? = In this case M (?; M ; ~a) has dimension ind(a1) + ind(a2) ? d, where ~a = (a1; a2). Thus q(?; M ) 2 q H q (M ) H d?q (M ), which de nes an element in q Hom(H q (M ); Hd?q (M )). This is the Poincare duality isomorphism, given by taking the cap product with the fundamental class. Example 3. ? = In this case M (?; M ; ~a) has dimension ind(a1) ? ind(a2) ? ind(a3), where ~a = (a1; a2; a3). Thus
q(?; M ) 2 rk H k (M ) Hr (M ) Hk?r (M ) and de nes an element in rk Hom(H r (M ) H k?r (M ); H k (M )): This is the cup product operation. Example 4. ? = 11
In this case M (?; M ; ~a) has dimension ind(a) ? d, where ~a = (a): Thus q(?; M ) 2 H d(M ). It is easily seen to be the Euler class (or Stiefel - Whitney class wd if M is not orientable). We end this section by discussing some basic structure properties of the invariants q(?; M ). (See [0] for details.) In particular the following three results say that the four examples above can be used to compute the invariant for any graph.
Proposition 7 If ?1 and ?2 are homotopy equivalent via a homotopy that preserves orientations on their end, then q(?1 ; M ) = q(?2; M ).
Now let ?1 and ?2 be oriented graphs. Let ?i1#;2j be the oriented graph obtained by gluing incoming edge i of ?1 to outgoing edge j of ?2.
Proposition 8 q(?i1#;2j ; M ) = q(?1; M ) i;j q(?2; M ); where i;j denotes tensorial contraction of cohomology in the ith coordinate with homology in the j th coordinate.
Corollary 9 Changing the orientation of a non-compact edge induces the Poincare duality isomorphism on the relevent tensor coordinate of the invariant q(?; M ).
12
2 The Equivariant Invariants In this section we will generalize the construction of section one to moduli spaces that are associated to families of M -structures. The invariants that we de ne will be elements of an equivariant (co)homology of the n-fold product of M . In basic cases we extract operations from H (M ) to H (M ) which are associated to elements of the homology of the orientation preserving symmetry group of ?, ?. In particular we will show how the classical Steenrod operations arise this way. The details of the results in this section can be found in [0]. Let M be a closed Riemannian manifold of dimension d and ? an oriented graph as in the last secton. Recall the projection onto the space of structures
: M(?; M ) ! S (?; M ): The group of automorphisms of the oriented graph ? acts naturally on both M(?; M ) and S (?; M ) and the map is equivariant. Notice that these actions are free because a structure associates distinct functions to di erent edges. Furthermore, since S (?; M ) is contractible, S (?; M )=? is a classifying space B ?. We will be considering the induced map on orbit spaces : M(?; M )=? ! S (?; M )=? ' B ?: We will be interested in the pull back under of certain families of structures. We choose to nd these families in smaller structure spaces which are easier to deal with. Let f : M ! R be a Morse-Smale function and let Uf C 1(M ) be a contractible neighborhood of f that contains only Morse-Smale functions. Consider the structure space Sf (?; M ) de ned by only allowing functions from this smaller set, Uf . The ? action restricts to Sf (?; M ) S (?; M ), so the quotient Sf (?; M )=? is well de ned. We note that the inclusion of Sf (?; M )=? in S (?; M )=? is a homotopy equivalence and both spaces are homotopy equivalent to B ?. Each Morse-Smale function g 2 Uf has a set of isolated critical points which we call Crit(g). Using a contraction of the neighborhood Uf one can de ne a xed bijective correspondence g : Crit(g) ! Crit(f ). 13
As in section one we distinguish elements of a moduli space, Mf (?; M ) = ? 1 (Sf (?; M )=? ), by the asymptotic behavior of each ?- ow along its ends.
To do this notice that the sets of critical points to which a ? - ow converge are permuted by the ? action. Given 2 Mf (?; M ), let ~a denote the orbit of the set of critical points. We call this the critical orbit of . Let
Mf (?; M ; ~a) Mf (?; M )=? be the subspace of those 2 Mf (?; M ) whose critical orbits correspond to
~a under the correspondence above. Now given a singular i-simplex, : i ! Sf (?; M )=? , de ne the pullback moduli space M (?; M ; ~a) = (Mf (?; M ; ~a)) which projects onto the standard i-simplex, i. The following are basic properties of these moduli spaces, and are analogous to ones proved in section one.
Theorem 10 For a generic singular i-simplex : i ! Sf (?; M )=? , the moduli space M (?; M ; ~a) is a manifold. The dimension of M (?; M ; ~a) is given by the formula
dim(M(?; M ; ~a)) = dim(M (?; M ; ~a)) + i: Furthermore, orientations of the manifold M and of the simplex i induce orientations on the moduli spaces M (?; M ; ~a).
Lemma 11 The zero dimensional moduli spaces M0 (?; M ; ~a) are compact, and thus nite.
Let #M0 (?; M ; ~a) be the number of points in the moduli space counted with Zor Z2 coecients. We must count with Z2 coecients when the moduli space is not oriented. We will use these moduli spaces to construct Q(?; M ), an element of H (E ? ? (M )n1 (M )n2 ), where ? acts on (M )n1 (M )n2 via the map ? ?! n1 n2 14
which assigns to a symmetry the associated permutation of the ends. First note that the inclusion of any nite subcomplex P of Sf (?; M )=? can be perturbed so that each simplex is generic in the sense of Theorem 1. This complex is covered by another complex P~ Sf (?; M ). Over Sf (?; M ), each ? - ow as associated to it (via the correspondence ) a well de ned n-tuple of critical points of f . We can then de ne the class QP (?; M ) 2 Hom? (C(P~ ); C (M; f ) n1 C(M; f ) n2 ) in the following manner. For each simplex in P~ consider those n - tuples of critical points ~a such that dim M(?; M ; ~a) = 0. These spaces contain a nite number of points which can be counted with sign (if M is not we count mod 2). Taking all these simplices together, we construct the desired element QP (?; M ).
De nition 4 QP (?; M )() =
X
#M0 (?; M ; ~a)[~a] 2 C (M; f ) n1 C(M; f ) n2 :
QP (?; M ) is clearly an equivariant homomorphism and so lies in Hom? (C(P~ ); C (M; f ) n1 C(M; f ) n2 ): This (tri) graded group is a cochain complex (i.e has a natural coboundary operator). The analogue of Lemma 2 in section one is the following.
Lemma 12 QP (?; M ) is a ? invariant cocycle. Thus it represents a cohomology class
QP (?; M ) 2 H (P~ ? (M )n1 (M )n2 ):
Since this construction is valid for every nite subcomplex of Sf (?; M )=? and is easily seen not to depend on the choices of perturbations, this process actually de nes a cohomology class 15
QP (?; M ) 2 H (Sf (?; M ) ? (M )n1 (M )n2 ) = H (E ? ? (M )n1 (M )n2 ): Notice that this class does not depend on the original function f . In order to produce cohomology operations from the class Q(?; M ) we apply it to elements in H(B ?).
De nition 5 For a class 2 H(B ?) de ne the invariant q (?; M ) to be the equivariant homology class,
q (?; M ) 2 H 1 ((M )n1 ) H 2 ((M )n2 ) given by evaluating Q(?; M ) on and using the homomorphism ? ! n1 n2 described above. n
n
Note that the same construction works for any subgroup G ? . The invariants of section one are those that arise when G is trivial. We now describe two examples of these equivariant invariants. Example 1. Let ? be the graph used in section one to obtain the Euler class. The symmetry group ? = Z2 which acts by exchanging the two internal edges. Since the group acts trivially on the noncompact end, the class Q(?; M ) takes its value in the Z2 cohomology of the product, B Z2 M . Thus for each generator
i 2 Hi(B Z2; Z2) = Z2
we have q (?; M ) 2 H d?i (M ; Z2). It is not dicult to see that this is the Stiefel-Whitney class !d?i (M ). Example 2. Let ? be the 3-ended graph used to obtain the cup product in section 1. Again the symmetry group ? = Z2 which acts by exchanging the two outgoing ends. In this case for the class i 2 Hi (B Z2; Z2) the invariant i
q (?; M ) 2 j HjZ2(M M ; Z2) H j?i (M ; Z2)
i
16
and so may be viewed as a homomorphism j (M M ; Z ) ?! H j ?i (M ; Z ): q (?; M ) : HZ 2 2 2 i
This is the cup ? i product. The Steenrod squaring operation Sqk?i occurs when precomposing this operation with the natural inclusion 2k (M M ; Z ) H k (M ; Z2) ?! HZ 2 2 ?! :
The composition of two Steenrod squares can be obtained here by considering a graph of the following type,
Here the symmetry group is the dihedral group D4. It is easy to see that the moduli space of this graph is homotopy equivalent to the moduli space of the following graph
This graph has larger symmetry group, 4. Thus the above dihedral group invariant factors through the symmetric group invariant. This produces the Adem relations among the Steenrod squares. A similar situation occurs when we compose the Steenrod squares with the cup product. In this case we need only consider the Z2 action on the rst vertex of the big graph. This gives the operations Sqi( [ ). Shrink the two internal edges and consider the larger symmetry Z2 Z2. The original symmetry maps into this one by the diagonal map. The kernel of the associated homology map is exactly the Cartan relations.
17
3 A Categorical Approach In this section we describe a more homotopy theoretic way of viewing the invariants. We describe categories CM(?;M ) and CE that have as their classifying spaces the moduli space M(?; M ) and the space of ends of M(?; M ) respectively. We then de ne an equivariant \ends" functor
E : CM(?;M ) ?! CE whose equivariant homological properties determine the invariants q(?; M ). The feature of studying the invariants this way is that it gives a clear method of calculation, and the homotopy nature of this de nition will allow for the de nition of higher order invariants de ned when the primary invariants q(?; M ) vanish. The details of the constructions in this section and the proofs of their properties will appear elsewhere in due course. We begin by recalling from [0] a categorical approach to Morse theory. The constructions in this section may be viewed as generalizations (or applications) of the results in [0]. Let f : M ! R be a smooth map on a closed, nite dimensional Riemannian manifold. Let Cf be the topological category whose objects are the critical points of f , and the morphisms between two critical points Mor(a; b) is given by Mor(a; b) = M(a; b); the compacti cation of the space of gradient ow lines given by \piecewise
ow lines" as discussed in section one. Let B Cf be the classifying space of this category. This is a standard construction which de nes a simplicial space which one k - simplex for every k -tuple of composable morphisms. The following is the main result of [0].
Theorem 13 There is a natural homotopy equivalence : B Cf ' M: For a generic f , (i.e a Morse function satisfying the Morse - Smale transversality conditions) is a homeomorphism.
18
This theorem will be useful when we are dealing with the moduli spaces with a xed structure, M (?; M ). However, as seen in the last section we often need to allow the structures to vary. For this we will need the following corollary of this theorem. Given a closed manifold M , let CM denote the topological category whose objects are pairs (f; a), where f : M ! R is a smooth function and a 2 M is a critical point of f . For morphisms we set (
; if f1 6= f2; Mor((f1 ; a1); (f2; a2)) = M ( a; b ) if f 1 = f2 f where Mf (a; b) is the space of piecewise gradient ow lines of the function f = f1 = f2. We then have the following result.
Corollary 14 There is a natural homotopy equivalence B CM ' M: Sketch of Proof. Let C 1(M ) denote the constant category whose ob-
jects are smooth functions on M . (By constant we mean a category whose only morphisms are the identity morphisms.) There is a clear functor : CM ! C 1(M )
de ned on objects by sending (f; a) to f . The ber category at any object f is Cf . By the theorem, on the classifying space level all these bers are homotopy equivalent to M . This will be the key fact in proving that is a quasi- bration of categories. That is, on the classifying space level : B CM ! C 1(M ) is a quasi- bration. Since C 1(M ) is contractible, the result will follow We will now use these results to study the moduli space M(?; M ) and the invariants q(?; M ) categorically. First, consider the compacti ed moduli space M(?; M ) described in section one. Recall that an element 2 M(?; M ) is a continuous map : ? ! M and a M - structure , so that when restricted to any internal edge is a gradient ow line, and restricted to any noncompact edge is a piecewise 19
ow with respect to the structure . Recall that we called such elements \piecewise ? - ows". Notice that when viewed simply as a set, there is a natural partial ordering on the elements of M(?; M ). Namely 1 < 2 if 2 is obtained from 1 by gluing on piecewise ows on the noncompact (incoming or outgoing) edges. Notice therefore that a minimal element in this partial ordering is any actual ? - ow
2 M(?; M ) M(?; M ): Now any partially order set X can be viewed as a category, where the objects are the elements in X . There exists a unique morphism between points x and y if and only if x y. If X is a topological space the resulting category may be viewed as a topological category. Let CM(?;M ) be the topological category associated to this partial ordering on the space M(?; M ). We then have the following.
Theorem 15 There is a natural homotopy equivalence between the classifying space of the category CM(?;M ) and the moduli space M(?; M ), B CM(?;M ) ' M(?; M ): Sketch of Proof. Given any partially ordered set X viewed as a category, we can consider the quotient category X= whose space of objects is X modulo the equivalence relation generated by the partial ordering. There are no morphisms other than the identity morphisms. Notice that there is a natural projection functor : X ?! X= : In the case of the category CM(?;M ) de ned from the partial ordering on M(?; M ), the quotient category M(?; M )= is homeomorphic to M(?; M ) (that is the space of objects is homeomorphic to M(?; M ) which can be viewed as a constant category). This is because every element in M(?; M )= is uniquely represented by an element 2 M(?; M ). This in turn is because every piecewise ? - ow ~ is to a unique ? - ow 2 M(?; M ). This 20
fact also implies that if one considers the ber category CM(?;M )( ) of the projection functor
: CM(?;M ) ?! M(?; M )= = M(?; M ) over any ? - ow , then CM(?;M )( ) is a partially ordered set with a unique minimal element. Therefore its classifying space is contractible. Then by applying Quillen's theory [0] one can conclude that the projection map : CM(?;M ) ! M(?; M ) induces a homotopy equivalence on classifying spaces
We are now ready to de ne our \ends functor" with which we will de ne the invariants q(?; M ). First we de ne the \end category" CE (?) as follows. Suppose the oriented graph ? has n1 incoming (noncompact) edges and n2 outgoing edges, where n1 + n2 = n. De ne the category CE as follows.
De nition 6 De ne CE (?) to be the product category Y Y CE (?) = CMop CM n2
n1
where CMop denotes the \opposite category", that is the category whose objects are the same as those of CM , but MorC (a; b) = MorC (b; a). op
We now de ne a functor
E : CM(?;M ) ?! CE (?) as follows. An object 2 CM(?;M ) is a piecewise ? - ow with respect to some M - structure 2 S (?; M ). As described in section one converges
to critical points of the functions labelling the incoming and outgoing edges that are given as part of the data of the structure . Thus determines an n tuple (f1; a1); ; (fn1 ; an1 ); (fn1+1 ; an1+1); ; (fn ; an) of labelling functions and critical points, where the rst n1 fi's label the incoming edges and the next n2 = n ? n1 label the outgoing edges. Notice that this n - tuple is an object in the category CE (?). Therefore on objects we de ne
E ( ) = (f1; a1); ; (fn1 ; an1 ); (fn1 +1; an1+1); ; (fn; an): 21
To de ne E on morphisms recall that in CM(?;M ) there is a unique morphism from 1 to 2 if and only if 2 1. That means there exists an n tuple of piecewise ow lines ( 1; ; n ) of the functions f1; ; fn respectively, so that 2 is obtained by gluing the i's onto the ends of 1. The rst n1 of the i's are glued with an incoming orientation and the next n2 are glued with an outgoing Qorientation. Notice that the n - tuple ( 1; n ) is Q op a morphism in CE (?) = n1 CM n2 CM . Therefore if ( 1; 2) denotes the unique morphism from 1 to 2 in CM(?;M ), we de ne
E (( 1; 2)) = ( 1; ; n ): As in section two let ? be the symmetry group of the oriented graph ?. Notice that ? acts freely on the category CM(?;M ). Furthermore the product of the symmetric groups n1 n2 acts freely on the end category CE (?). The following is a straightforward check of de nitions.
Proposition 16 The map E : CM(?;M ) ?! CE (?) is a well de ned equivariant functor with respect to the homomorphism ? ! n1 n2 given by sending a symmetry to the associated permutation of the ends.
To de ne the invariants q(?; M ) in this context we will be studying the map induced by E in equivariant homology. Speci cally, consider the composition
Q : Hk? (B CM(?;M )) = Hk? (M(?; M )) ! Hk 1 2 (B CE (?)) = k1+k2 =k Hk1 1 (Qn1 B CMop) Hk2 2 (Qn2 B CM ) = k1+k2 =k Hdn11?k1 ((M )n1 ) Hk2 2 ((M )n2 ) ! k1+k2 =k Hom Hk2 2 ((M )n2 ); Hdn11?k1 ((M )n1 ) : n
n
n
n
n
n
n
n
In the previous section we discussed how to de ne an invariant q (?; M ) given any element 2 H(B ?). Here will describe how q (?; M ) is simply 22
the image under the above map Q of an equivariant homology class ( ) 2 H? (M(?; M )). The rst step is to study the relation of the equivariant moduli space to the space of ends equivariantly. For this consider the map to the structure space : M(?; M ) ?! S (?; M ) studied previously. By abuse of notation we will let S (?; M ) also denote the constant topological category whose objects are the space of structures. Then the map can be realized on the categorical level : CM(?;M ) ?! S (?; M ) which is ? - equivariant. Consider the induced functor on orbit categories, : CM(?;M )=? ?! S (?; M )=? ' B ?: The following result is the main technical result, but is proved by relatively straightforward transversality arguments.
Theorem 17 Let N S (?; M )=? ' B ? be a closed submanifold of dimension p. Let CM(?;M )(N ) = ?1 (N ) CM(?;M )=? be the subcategory of CM(?;M )=? whose objects have structures lying in N . Then for a generic embedding of N in S (?; M )=? the classifying space B CM(?;M )(N ) has the
homotopy type of a closed manifold of dimension
dim B CM(?;M )(N ) = p ? (r ? 1)d where, as before, r = rankH1 (?) and d = dim M . We use this result as follows. Let N ,! S (?; M ) ' B ? be a closed manifold of dimension p representing 2 Hp (B ?). Let ( ) 2 Hp??(r?1)d(M(?; M )) be the homology represented by the inclusion B CM(?;M )(N ) B CM(?;M ) ' M(?; M ) given by the above theorem. Consider the class
Q(( )) 2 k Hom(Hk 2 ((M )n2 ); Hk+1d(n1 +r?1)?p ((M )n1 )) n
n
23
de ned as the image of ( ) under the composite map Q de ned above. The following is the main result of this section, and is proved using standard techniques that describe intersections in terms of homology.
Theorem 18 Given any 2 H(B ?), Q(( )) = q (?; M ): The main e ect of this result is that the invariants q (?; M ) can be de ned globally in terms of equivariant homology. In particular one does not need to perturb metrics or functions to defne these invariants. This may have calculational advantages. We end this paper by describing how, in practice, these invariants may be computed. Any oriented graph ? is homotopy equivalent relative to its (oriented) ends to a graph ?0 with a single vertex v0, with n1 incoming noncompact edges, n2 outgoing noncompact edges, and r loops (internal edges which both start and end at vo) for some n1, n2, r = rank H1 (?). The symmetry goup of ?0 is given by ? = n1 n2 r : Via this homotopy equivalence an element 2 H(B ?) determines an element 0 2 H(B (n1 n2 r )): 0
Now by pinching the r looped internal edges in the graph ?0 to a point, one gets a projection map to the simply simply connected graph ?0 that has a single vertex v0, n1 incoming noncompact edges, n2 outgoing noncompact edges, and no internal edges. The map of graphs ? ' ?0 ! ?0 induces a homomorphism of their symmetry groups ? ! ?0 = n1 n2 which is the homomorphism described above given by sending a symmetry of ? to the induced permutation of the ends. Thus 2 H(B ?) is mapped to a class 0 2 H(B (n1 n2 )). Also the class ( ) 2 H? (M(?; M )) is mapped to 0 2 H 1 2 (M(?0; M )) n
n
24
. By the naturality of the construction we see that q (?; M ) is given by
Q(0) 2 Hom(H 2 ((M )n2 ); H 1 ((M )n1 )) n
n
where Q here is the homomorphism in equivariant homology induced by the end functor
E : B CM(?0;M ) ! B CE (?) = (B CMop)n1 (B CM )n2 as above. Thus what remains is to study the equivariant homotopy type of this map. This is done as follows.
Theorem 19 There is a natural equivariant homeomorphism h : M(?0; M ) = S (?0 ; m) M de ned by sending to (; (v0)) where is the structure associated to . Furthermore, after passing to orbit spaces the realization of the end functor is homotopy equivalent to the equivariant diagonal map:
: B (n1 n2 ) M ?! E n1 1 (M )n1 E n2 2 (M )n2 : n
n
Notice that this theorem and the above observation says that calculation of an invariant q (?; M ) reduces to computing the above equivariant diagonal maps in homology. These can be viewed as generalized cupi products and their calculations are equivalent to the calculation in homology of groups of the inclusion of products of symmetric groups into larger symmetric groups. These calculations are well known. Notice also that since the invariants q (?; M ) can be de ned entirely in terms of the equivariant properties of the induced map on classifying spaces of the end functor B E , then by taking the homotopy ber of this map we can de ne functional and higher order operations. We will return to this topic in another paper.
25
References [1] M. Betz, Categorical constructions in Morse theory and cohomology operations Stanford Univ. Ph.D thesis, 1993. [2] R.L. Cohen, J.D.S. Jones, and G.B. Segal, Morse theory and classifying spaces Stanford University preprint, 1991. [3] R.L. Cohen, J.D.S. Jones, and G.B. Segal, Floer's in nite dimensional Morse theory and homotopy theory to appear, Proceedings of Floer Memorial Conference, 1993. [4] S.K. Donaldson, Polynomial invariants for smooth four- manifolds Topology vol. 29 no. 3 (1990), 257-316. [5] S.K. Donaldson and M. Furuta Floer Homology Theory, to appear. [6] D. Quillen Higher algebraic K - theory I Springer Lect. Notes vol. 341 (1973), 77-139. [7] Y. Ruan Symplectic topology on algebraic 3-folds to appear, Proceedings of Floer Memorial Conference, 1993. [8] E. Witten Topological quantum eld theory Comm. Math. Physics vol. 117 (1988), 353-386. [9] E. Witten Two dimensional gravity and intersection theory on moduli spaces Surveys in Di . Geo. vol. 1 (1991), 243-310.
26
