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A lift of the Chern-Simons functional and its application to equivariant Floer homology Anderson, Vaughn 1995

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A LIFT OF THE CHERN-SIMONS FUNCTIONAL AND ITSAPPLICATION TO EQUIVARIANT FLOER HOMOLOGYByVaughn AndersonB. Sc. (Mathematics) University of SaskatchewanA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYINTHE FACULTY OF GRADUATE STUDIESMATHEMATICSWE ACCEPT THIS THESIS AS CONFORMINGTO THE REQUIRED STANDARDTHE UNIVERSITY OF BRITISH COLUMBIAOCTOBER 1995© VAUGHN ANDERSON, 1995In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall ot be allowed without my written permission.MathematicsThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1Z1Date: 1nJAbstractWe investigate the gauge theory of 3- and 4- manifolds. A lift of the Chern-Simonsfunctional for fiat connections on a principal SU(2)-bundle over a homology 3-sphereY is constructed, putting strong restrictions on the existence of low-dimensional instanton moduli spaces over the cylinder 1k x Y. The value of this lift is computed for theBrieskorn spheres E(p, q, pqk — 1), and there is found to be one and only one fiat connection of Floer index 1 with positive Chern-Simons functional. This fact is applied tothe computation of the equivariant Floer homology HFq,(E(2, 3, 6k — 1)) showing thatthere are connections of index 1 and 5 with non-trivial boundary in the equivariant Floerhomology. Specializing to the case k = 2, we obtain vanishing ofHF0,4+1((2,3, 11)).IITable of ContentsAbstract iiTable of Contents iiiList of Figures vGlossary of Notation viAcknowledgment ix1 Introduction 12 Background 102.1 Donaidson-Floer Theory 102.1.1 The Space of Connections 102.1.2 The Donaldson Polynomial 132.1.3 The Chern-Simons Functional cs and Floer Homology HF(Y) . 192.1.4 Gluing of Donaldson Polynomials 262.2 Equivariant Floer Theory 282.2.1 Equivariant Homology 292.2.2 Morse-Bott Theory and Equivariant Floer Theory 302.2.3 The Relative Donaldson Polynomial for Equivariant Floer Homology 363 The Chern-Simons Functional 393.1 A Real-Valued Lift of the Chern-Simons Functional 391113.2 Calculations on Brieskorn Homology Spheres 413.2.1 Flat Connections over E(p,q,pqk— 1) 423.2.2 Index Calculations 453.2.3 A Geometric Interpretation of W(a, e) 463.2.4 A Distinguished Connection with Index 1 553.2.5 Monotonicity of the Chern-Simons Functional 664 Computing Donaldson Polynomials using Equivariant Floer Theory 684.1 The Canonical Resolution of the (2, 3, 6k — 1)-singularity 694.2 Superfluous Elements of Floer Homology 724.2.1 Computing d1 744.2.2 Frøyshov’s Invariant and the Elements of HF5((2,3, 6k — 1)) 764.2.3 The Image of the Donaldson Polynomials 794.3 Applications to (2,3,11) 80Bibliography 83ivList of Figures2.1 The spectral flow of a path in A 232.2 Spectral sequence for HFG,(E(2,3,6k — 1)) 343.3 The orbit of a connection a E A under the action of G. 403.4 Sphere S2 with lines through fixed points of X, Y, and Z 433.5 A tetrahedron in the unit cube 443.6 A count of lattice points inside the parallelogram 503.7 Reducing Z(x,y) to (x,y’) 533.8 The triangle on which z(x, y) is defined 533.9 A new basis for the integer lattice 543.10 A count of lattice points along the x-axis 553.11 Reduction of a count of lattice points to a small wedge 603.12 The symmetry in the count (e/q, e/p) 623.13 Estimation of 5(e/q, e/p) 634.14 The surgery diagram for the canonical resolution of (2, 3, 6k — 1) 704.15 The spectral sequence Ep’q for HFG,() 73VGlossary of Notationri(X), 1 , 12Ix,2 ,12H2(X), 2 *, 12u(X), 2 [L, 138X1, 4 1, 13Ay/cy, 6 pi(E), 13bt(X), 6 *, 13F,Fk, 11 1(X), 14c2(P), 11 P, 14TP, 11 F, 14GL(V), 11 Mk, 14A,Ak, 11 YM, 141(X,Adzu(2)), 11 d(k), 15FA, 11 b1, 15c,ck, 11 R, 15S’, 11 w2(R), 15Q, 11 J4pi(R),w2(R , 15g, ii d(pi(R),w(R) , 15A*, 11 Dx, 15B, 11 S*(H2(X;Z)), 15L, 12 IMk, 16ci(L), 12 s’(X), 17pA 12 V, 17B*, 12 A(X), 18viDx, 18K, 19cs, 20dcs, 20tr, 20R.y, 21A, 21*3, 21M(a,/3), 22i(a,3), 22*dA, 22SF(a, 8, I’), 23SF(a,3), 231(a), 24HF(Y), 24(CF(Y),‘9Floer), 24aFloer, 24(CF*(Y), dFloer), 25HF*(Y), 251(a), 25,26Dx1, 26M(X1,p), 26MG, 29EG, 29BG, 29HG,(M; R), 2930C(M), 30S*(g), 30300a 31CFc,*(Y), 31u,l, 31HFç,*(Y), 320, 32HFJ(Y), 32(C(M),G), 32(CF’(Y),), 3233vol, 333611, 37_bx1, 3740(ai,a2,3) 41a, 42b, 42b0, 42l,m,n, 44e, 45i, 46viiR(e), 46W(a, e), 466, 488(x,y), 51(x,y), 51(x,y), 521, 55S(e), 564,77M(2,3,11), 81viiiAcknowledgmentI would like to thank my supervisor, David Austin, for suggesting this problem to me,and for his support and encouragement. I would also like to thank the Mathematicsdepartment of the University of British Columbia for its hospitality during my studies inVancouver.Additionally, I would like to thank Ben Nasatyr, Weiping Li, and David Auckly forhelpful conversations.I am indebted to the National Science and Engineering Research Council for theirsupport during my tenure at UBC, and encourage the government of Canada’s continuedsupport of fundamental research.ixChapter 1IntroductionIn the early 1980’s, Simon Donaldson revolutionalized our understanding of smooth four-dimensional topology by introducing new techniques into the field and using them toproduce dramatic new results. Four-dimensional topology possesses a peculiar nature—it lies on the boundary between what we can visualize (dimensions three and lower)and what we can effectively manipulate (dimensions five and higher). Consequently,deep questions about four-manifolds remained unapproachable to mathematicians beforeDonaldson’s work. In this thesis, we will present some results concerning the computationof Donaldson’s polynomial invariants. First, however, we will give some background inthis introduction which will put these results into context.A fundamental problem in the area is to provide a classification of 4-dimensionalmanifolds. However, this turns out to be an impossible task: for any finitely presentedgroup G, there is a smooth 4-manifold X whose fundamental group 7r1(X) G. Sinceit is known that there is no general algorithm for distinguishing two finitely presentedgroups, there can be no general algorithm for distinguishing general 4-manifolds from oneanother. For this reason, we will restrict our attention to the case of simply connectedmanifolds.Perhaps the first result in the classification of 4-manifolds is due to J.H.C. Whiteheadwho showed that two simply-connected oriented topological 4-manifolds are homotopyequivalent if and only if their intersection forms are isomorphic [48]. The intersection1Chapter 1. Introduction 2form 1x on a 4-manifold X is a symmetric, unimodular bilinear form(1.1) Ix : H2(X; Z) ® H2(X; Z) — Z.which may be thought of as an oriented count of the intersection points of surfaces in X.Whitehead’s theorem says that the intersection form completely determines the homotopy type of a simply-connected topological 4-manifold and points to the pivotal roleplayed by the intersection form in 4-manifold topology and hence in this thesis. Someimportant quantities are associated to the intersection form. First is the rank of theform, denoted by b2(X). Moreover, since the intersection form is symmetric, it may bediagonalized over the reals (though not necessarily over the integers). We will denotethe number of positive and negative eigenvalues in such a diagonalization by b(X) andb(X). Since Ix is unimodular, it follows that b2(X) = b(X) + b(X). The signature ofIx is defined to be a(X) = bt(X) — b(X).In 1982, two amazing results appeared concerning the structure of 4-manifolds. Thefirst was M. Freedman’s classification of topological 4-manifolds [31]. Freedman showedthat the intersection form was an almost complete invariant of topological 4-manifolds;that is, every unimodular, symmetric, bilinear form is realized as the intersection formof a 4-manifold and at most two distinct topological 4-manifolds realize such a givenform. Freedman’s result involved subtly applying the techniques which are successful forstudying higher dimensional manifolds to the problem in dimension four. A corollary ofthis result is the topological Poincaré conjecture in dimension four.The second result, and the one most relevant to this thesis, was Donaldson’s firsttheorem [13] about the intersection forms realized by smooth 4-manifolds. Donaldsonshowed that the only intersection forms realized by a smooth manifold with a definiteintersection form are the diagonalizable ones. This result was an incredibly hopefuloutcome for the classification of smooth 4-manifolds since the number of positive definiteChapter 1. Introduction 3forms grows quickly with the rank, see [43]; for example, the set of bilinear forms ofrank 32 has over iO distinct equivalence classes under isomorphism! Moreover, thedefinite forms are the ones which have resisted a classification scheme. Together withFreedman’s theorem, this produces a large number of topological 4-manifolds with nosmooth structure. The techniques involved in Donaldson’s theorem utilize solutions topartial differential equations—called instantons—arising in mathematical physics. Thisstudy is loosely known as gauge theory.Combining Freedman and Donaldson’s results gives the surprising corollary statingthe existence of many different smooth structures on R4. Dimension four is the onlydimension in which Euclidean space has such exotic smooth structures.Donaldson’s first theorem addresses the existence of smooth structures on topological 4-manifolds. Later Donaldson introduced his polynomial invariants [18] which heused to demonstrate the non-uniqueness of smooth structures on (closed) topological 4-manifolds. One consequence is that the h-cobordism theorem, the principle tool in higherdimensional topology, does not hold in dimension four. For a simply connected closed4-manifold, Donaldson’s polynomials are essentially defined as a count of the numberof connections which satisfy the anti-self-duality equations and which obey some constraints. These polynomials are able to distinguish between smooth structures on manyhomeomorphic algebraic surfaces.For algebraic surfaces, Donaldson [16] found a correspondence between the anti-self-dual connections and stable holomorphic vector bundles over the surface. This correspondence may be used to show that the Donaldson polynomial for an algebraic surfaceis non-zero.The Donaldson polynomials are elements in a (restricted) topological quantum fieldtheory(TQFT); that is, they may be computed through a cut-and-paste method similarto the Mayer-Vietoris exact sequence for computing homology. To be more specific, thisChapter 1. Introduction 4means that for a manifold Y in a large class of 3-manifolds (including the homology 3-spheres), there is an associated Z/8Z-graded vector space known as the Floer homologyHF(Y). This space was introduced by A. Floer [29] and is defined using the “Morsecomplex” for the Chern-Simons functional. This functional cs : By —* R/4Z is definedon By, the space of connections on the trivial SU(2)-bundle over Y modulo gauge equivalence. Formally, one may think of HF(Y) as the homology of By. Letting Y be Y withthe opposite orientation we have a pairing(1.2) (,) : HF(Y) 0 HFj —*The importance of Floer homology in computing Donaldson polynomials is the following. To a smooth, oriented, compact 4-manifold X1 with boundary oX1 = Y, there isa polynomial Dx, : S*(H2(Xi; Z)) —* HF(Y). Additionally, if X2 is a 4-manifold withOX2 = Y then by gluing X1 and X2 along their common boundary Y we obtain theclosed manifold(1.3) X = (X1ll2)/’ , yid(y)(1.4) = X1uX2where id: Y — is the orientation reversing diffeomorphism given by the identity. Under some additional constraints, the Donaldson polynomial may be expressed by pairingthe relative Donaldson polynomials:(1.5) (Dx1,2) Dx.Ideally by dissecting a manifold into sufficiently simple pieces one could calculate theDonaldson polynomial. In practice, this has been an effective technique; for example,Fintushel-Stern [28] and Lisca [42] have computed the Donaldson polynomials for severalclasses of 4-manifolds by decomposing along a simple homology 3-sphere.Chapter 1. Introduction 5A second application of Floer homology is to produce vanishing theorems. Noticethat if the Floer homology of Y is trivial—that is, HF(Y) = 0—then the Donaldsonpolynomial for any 4-manifold X with Y embedded in X in such a way that (1.5) holdsmust necessarily be zero. This allows one to obtain results on the possibility of embeddingcertain 3-manifolds into smooth 4-manifolds. To be specific, when there is an embeddingwhich decomposes X = X1 UyX2 along a homology 3-sphere Y with bt(X) > 0, i = 1,2,the Donaldson polynomial factors through the Floer homology as in equation (1.5). Thefirst such vanishing theorem, which actually predated the introduction of Floer homology,is due to Donaldson [16]: If X = X1#X2is an algebraic surface then b(X) = 0 for eitheri 1 or 2. A corollary of this theorem is the indecomposability of algebraic surfaces—thatis, an algebraic surface is not the smooth connected sum of any two algebraic surfaces.This follows since b(X) > 0 for any algebraic surface X.More generally, J. Morgan, T. Mrowka, and D. Ruberman [44] have proven that theDonaldson polynomial for 4-manifolds split along a lens space such that b(X) > 0must vanish. This has the interesting consequence that the Donaldson polynomial ofa 4-manifold vanishes if it contains an embedded 2-sphere of positive self-intersection,and hence algebraic surfaces have no smoothly embedded 2-spheres of positive self-intersection. This demonstrates that gauge theory has implications for the study ofsmooth embeddings of 2-manifolds into a 4-manifold.One of the complications in defining Floer homology is the fact that the space By isnot a manifold. The space By is the quotient of the space of connections Ày under theaction of the natural group of symmetries—the gauge group y. When Y is a homology3-sphere, there is a singular point corresponding to the trivial connection 8 which has nontrivial isotropy under the action of y. Floer addressed this problem by simply excisingthe trivial connection.However, another possible remedy is to only consider the action of the so-called framedChapter 1. Introduction 6gauge group g C y. This group acts freely on Ày, and we arrive at the space offramed connections B = A/c. The SO(3)-action on y induced by the action ofcic the isotropy group of 0 has quotient By = B/SO(3). Now D.M. Austin andP.J. Braam [5] have developed a modification of Floer homology, the equivariant Floerhomology HFc,*(Y), obtained by formally computing the SO(3)-equivariant homologyof the SO(3)-space y.The space HFç,(Y) is also a Z/8Z-graded vector space but possesses additionalstructure. For instance, the space HFc,*(Y) is a module over the SO(3)-equivariant cohomology of a point H*(BSO(3); IR) JR[u], where u is a generator of degree 4. Thismodule structure gives an interpretation of the so-called “u”-map which has been previously studied in [8].This equivariant Floer group generalizes the usual Donaidson-Floer picture outlinedabove. In particular, a gluing formula comparable to (1.5) holds in greater generality.For instance, a relative polynomial-Ox1 : S*(H2(Xi;Z)) —* HFç,,(Y) is defined withvalues in equivariant Floer homology. Moreover, there is a pairing of the equivariantFloer groups which expresses the Donaldson polynomial through the relative polynomialsDx = (x1,Dx2).For a homology sphere, the two forms of Floer homology are related through an exacttriangle [5].HF(Y)g(1.6) HFc,*(Y).The composition of the map HFo,(Y) —* HF(Y) with 0x1 : S*(H2(Xi;)) —* HFg,*(Y)induces the relative Donaldson polynomial Dx1 : S*(H2(Xi; Z)) —* HF(Y).Chapter 1. Introduction 7By computing the equivariant Floer homology of the 3-manifolds S3/F, where F isa finite subgroup of SU(2), Austin [4] has proven that the Donaldson polynomial for a4-manifold split by S3/F vanish provided bt(X) > 0 (there are a few exceptions). Thisagain implies that algebraic surfaces cannot be split by S3/F unless b (Xi) = 0 for i = 1or 2.The latest development came from N. Seiberg and E. Witten [49]; by studyingmonopoles on 4-manifolds, they found a generalization of the above theorems of Donaldson, Morgan-Mrowka-Ruberman and Austin. One consequence is that algebraic surfacesmay not be split by a 3-manifold admitting a metric with positive curvature unless14(X) =0 for i = 1 or 2.Once the Donaidson-Floer picture is in place, it is natural to ask if the entire Floerhomology HF(Y) is necessary to express the computation of Donaldson polynomialsthrough this cut-and-paste procedure. For instance, if there was a smaller group contained in HF(Y) which was guaranteed to contain the relative Donaldson polynomials,this would give relations between the various Donaldson polynomials and perhaps vanishing in certain cases. One result in this direction is due to P.Lisca [42] who has shownthat the relative Donaldson polynomials of the Gompf nuclei generate the Floer homologyHF4+3(E(2, 3, 6k — 1)). This means, that the entire Floer group HF4+3((2,3, 6k — 1))is necessary for the computation of the Donaldson polynomial and then uses the blow-upformulae of R. Friedman and J. Morgan [32] to compute a specific relative Donaldsonpolynomial on the Gompf nucleus. Lisca uses these calculations in the completion of theclassification of minimal elliptic surfaces: two elliptic surfaces of genus n — 1 and twomultiple fibres, and P’’ are difFeomorphic if and only if p = p’ and q = q’.In this thesis, we partially compute the equivariant Floer homology for a class ofBrieskorn homology spheres E(2, 3,6k — 1). As a corollary of this computation we obtainthe following theorem.Chapter 1. Introduction 8Theorem 1.1 There is a subspace V C HF4+1((2,3,6k — 1)) of at least co-dimensiontwo such that the following holds. If Xi is a compact, simply-connected, smooth, oriented4-manifold with OX1 = (2, 3,6k — 1), then the component of the Donaldson polynomialDx1 : S*(H2(Xi;Z)) —*HF4+1(E(2,3,6k —1)) lies in V. That is,S*(H2(Xi;Z))—* V CHF4+1(E(2,3,6k —1)).When k = 2, this may be strengthened to the following corollaryCorollary 1.2 If X1 is a compact, simply-connected, smooth, oriented 4-manifold withOX1 = (2, 3, 11), the component of the Donaldson polynomial Dx1 : S*(H2(Xi; Z))HF4+((2,3,6k —1)) is zero.This should be thought of as an analogue of the vanishing results quoted above. Unfortunately, this does not seem to be strong enough to force the vanishing of Donaldsonpolynomial for any 4-manifolds split by (2, 3, 11).Besides these results, the techniques used are novel and may be of independent interest. Our principal tool is the relationship between the monotonicity properties of theChern-Simons functional and the existence of instantons on the cylinder R x Y. Withthis in mind, we construct a lift of the Chern-Simons functional to the reals for fiat connections on the principal SU(2)-bundle over Y. The lift gives strong restrictions on theexistence of gradient flow lines which are precisely instantons on the cylinder. The valueof the (non-lifted) cs : B — R/Z for flat connections have been calculated in a variety ofways by Fintushel and Stern [25] for Brieskorn spheres and P. Kirk and E. Klassen [37],and D. Auckly [3] for manifolds fibred over S1. By careful examination of the computations of Fintushel and Stern [26], we develop a combinatorial method which computesthe lift of cs for Brieskorn homology spheres E(p, q, pqk — 1). Specifically, we have thefollowingChapter 1. Introduction 9Theorem 1.3 There is a unique flat connectiont E HF1(E(p,q,pqk— 1)) for which thelift is positive.This connection i. is also the connection which has Chern-Simons value is closest to 0modulo Z, as found by Fintushel and Stern [25]. We will then see how this leads to thecorollaryCorollary 1.4 There is only one nonempty one-dimensional moduli space of instantonsM(cr,O) on the cylinder R x E(p,q,pqk —1).Our lift of cs to flat connections is constructed by fixing the spectral flow between flatconnections. This is related to a construction of Fintushel and Stern [271 who constructa different lift by constraining the value of cs to lie between 0 and 1. This restriction isused to define integer-graded Floer homology groups.We apply the uniqueness of the connection i to the calculation of the equivariant Floerhomology of (2, 3, 6k — 1). Using the exact triangle (1.6), define the subspace V to bethe image of the map g. We will first show that i V and so HF1(E(2, 3,6k— 1))/V 0.Combining this information with computations of K.A. Frøyshov [33], we show that thereis a class v e HF5(E(2, 3, 6k — 1)) such that v V. This directly leads to the resultsmentioned above.In Chapter 2, we will provide the basic background for the Donaldson-Floer computations. In particular, we will carefully present the technical material outlined in thisintroduction. Chapter 3 gives the definition and computation of the lift of the ChernSimons function and the proofs of Theorem 1.3 and Corollary 1.4. Then in Chapter 4we apply Frøyshov’s computations and our work in Chapter 3 to prove Theorem 1.1 andCorollary 1.2.Chapter 2BackgroundIn this chapter, we will outline the differential geometry underlying Donaidson-Floer theory. We will then review the traditional Donaidson-Floer invariants and a generalizationwhich will be significant for our purposes.In this chapter all 4-manifolds are assumed to be smooth, oriented, and simply connected. Unless a boundary is explicitly given, we will assume our 4-manifolds to beclosed.2.1 Donaldson-Floer TheoryIn this section, we will review the construction of the Donaldson polynomial, Dx, for asmooth, closed, oriented, simply-connected 4-manifold X and cut-and-paste techniquesfor its computation. This leads naturally to the Floer homology HF(Y) of a homology3-sphere Y and to relative Donaldson polynomials. References for this material may befound in Donaldson-Kronheimer [20], Donaidson-Furuta [19] and Donaldson [15].2.1.1 The Space of ConnectionsHere we will introduce the space of connections on a principal SU(2)-bundle over a 4-manifold and the action of a natural group of symmetries. This material is standarddifferential geometry and may be found in the references Kobayashi-Nomizu [38] andLawson [41].The orientation of X gives a natural isomorphism H4(X; Z) 7Z. Now to each k E Z,10Chapter 2. Background 11there is a principal SU(2)-bnndle it : Fk —* X whose second Chern class c2(Fk) = k.When no confusion can occur, we will drop the subscript k from our notation.A connection A on F is commonly described in two equivalent ways: as a smoothsplitting TF zu(2) e ir*TX; or as a covariant derivative on an associated vector bundleE = F x,, V, where p: SU(2) —, GL(V) is a representation of SU(2) on the vector spaceV. The set of all connections forms a topological space, denoted Ak, which is an affinespace modeled on Q’(X, Adsu(2)). Here Adzu(2) is the bundle associated to F throughthe adjoint representation of SU(2).A connection A has an associated quantity called the curvature usually denoted byFA C Q2(X, Adzu(2)). The curvature is the derivative of the connection A when interpreted in the appropriate sense.We will denote the automorphism group of F by = Aut(F) and call it the gaugegroup of the bundle. This group acts on A in a natural way. If w C Q’(F,su(2)) is theprojection onto the vertical bundle determined by a connection A, a gauge transformationg C produces a projection g*w C Il’ (F, p) corresponding to the action of g on A. Thisaction is not free; indeed, many of the complications in gauge theory arise from the factthat the 0-action on A has varying isotropy. We will now discuss this more fully.First of all, there is a Z/2Z subgroup of 0 which fixes every connection. Secondly, ifthere is a principal S1-bundle Q such that F = Q Xsi SU(2), then connections whicharise as connections on Q are fixed by an S’-subgroup of 0. Lastly, for the special casewhen F is the trivial bundle, a product connection 0 is fixed by an SU(2)-subgroup of.We then define the group Q to be modulo the centre, = 0/(Z/2Z), and call aconnection irreducible if acts freely upon it. Let A* be the irreducible connections inA and 8* = A*/g. If a connection is not irreducible it is said to be reducible.Because the action of is not free, singularities arise in the quotient space B = A/Q =A/ due to the presence of reducible connections. We will see that these singularitiesChapter 2. Background 12form a relatively small subset of B—in particular, they are a set of infinite codimension.For instance, at an S’-reducible A E Ak, the vector bundle, E, associated to Pthrough the canonical representation, splits as E = L L*, where the first Chern classci(L) satisfies ci(L)2 —k. Then the tangent space to A splits as V(X,Adzu(2)) =Z’(X, iR) f’(X, L2). On ‘(X, iR) the action of the isotropy group = S1 is trivial,but for f(X, L2), S’ acts by multiplication— e6 = e26. Thus the link of areducible A e Bk is then homotopy equivalent to CP°°.The difficulties encountered because of the singular points of B can be dealt with intwo different ways. First, the singular points can simply be excised; in this case we lookat only the space B*. In general, for a subset S C A (or B), we define S* S fl A* (orSflB*).A second way to avoid the presence of singular points is to use a smaller group ofsymmetries. For a fixed point x X, define the framed gauge group as(2.1) g = {g E g(x) = id}.This group acts freely on the space of connections A and so this larger quotient space= A/ of framed connections is a manifold. The presence of reducible connections isnow detected through the action of the quotient group(2.2) 50(3).The action of 50(3) on has quotient(2.3) B = 5/50(3).Restricting attention to the irreducibles, the space is the total space of a principalS0(3)-bundle over B*. This fibration is called the base-point fibration.Chapter 2. Background 13We now define a map 1u: H2(X;) —f H (13*) in terms of a principal SO(3)-bundleE —* X x 13*• Here E is defined as the quotient(2.4) II = (P x .A*)/(2.5) = PxA*,where the equivalence relation identifies (p, A) c’-’ (pg(p), g’A) for g g. The map isdefined using the slant product with the first Pontrjagin class pi(E) E H4(X x 13*; Z),(2.6) p(E) =pi(Th)/>.2.1.2 The Donaldson PolynomialIn this section, we will introduce the Donaldson polynomial as defined by Donaldson[18] and its refinement, the Donaldson series, defined by Kronheimer-Mrowka [39]. Thisconstruction is special to smooth 4-dimensional manifolds. What makes this dimensionunique is the splitting of 2-forms into self-dual and anti-self-dual parts givell by a metric.Though the construction of the Donaldson polynomial depends on the Riemannian metric, the polynomial is independent of the metric, depending only on the smooth structureover the 4-manifold.Associated to an oriented smooth Riemannian manifold M of dimension n is theHodge-* operator, a linear map(2.7) * : f(M) —÷depending on the orientation and the metric. For a given metric on the 4-manifold X,the map(2.8) * : Q2(X)—f2(X)Chapter 2. Background 14defines an involution on Q2(X). The decomposition of the 2-forms 1Z2(X) into the +1-eigenspaces of *, f(X), splits the 2-forms as(2.9) V(X) = X) 1(X).Elements of the space X),(respective1y, Q. (X)) are known as self-dual (respectively,anti-self-dual) 2-forms. This splitting extends to vector-valued 2-forms,(2.10) V(X, Adzu(2)) = Adsu(2)) Adzu(2)).Denote by P÷ the projection to 1(X,Adu(2)).In particular, we will consider the self-dual part of the curvature = P÷FA. Aconnection A with F = 0 is called anti-self-dual and we define the moduli space ofinstantons to be(2.11) {A E = o}ic.We will often abbreviate anti-self-dual as ASD. Let M = Mk fl B consist of theirreducible ASD connections.It is important to note that the projection P depends on the metric. Varying thechoice of the metric changes the instanton moduli space. To arrive at topological information, we need to ensure that the information extracted from Mk is invariant of thechoice of metric.The instanton moduli space arises in theoretical physics in the following way. Physicists are interested in the gauge-invariant Yang-Mills functional YM : Ak —* 11 givenby(2.12) YM(A) =_--j tr(FA A *FA).The gauge equivalence classes of the absolute minima of this functional are precisely theinstantons.Chapter 2. Background 15For a generic choice of metric, M is a smooth manifold whose dimension, 2d(k), isgiven by the Atiyah-Singer index formula [30, §3]. Let b1 be the rank of Hi(X; IR) andb the rank of a maximal positive definite subspace of H2(X; IR) with respect to Ix Thedimeilsion 2d(k) of the moduli space is(2.13) 2d(k) = 8k— 3(1 — b1(X) + b(X)).In case X is simply connected, b1(X) = 0 and the dimension of Mk is even if and onlyif b(X) is odd.Often it will be useful to work with instantons defined on a principal SO(3)-bundleR —* X. Such bundles are classified by their first Pontrjagin class pi (R) and secondSteifel-Whitney class w2(R). The formal dimension 2d(pi (R),w2(R)) of the moduli space— I A ,— 4* z’A— n1 ii’./VLP1(R)w2)— p,(R),w2( ) L + — VJ/I iS(2.14) 2d(pi(R),w(R)) = —2p1(R) —3(1 — b1 +b(X)).Donaldson [17] shows that the instanton moduli spaces Mk are all orientable. In fact,given a homology orientation—that is, an orientation of the vector space H°(X; R)H(X; R)— the moduli spaces acquire a canonical orientation.Ideally it would be possible to define the Donaldson polynomial Dx : Sd(H2(X; Z))Z for d = d(k) an integer(i.e. b(X) odd) as the following integral: for x1,... , xd EH2(X;Z),set(2.15) Dx(x1,. . .,Xd) = ((x1) . . . IL(xd), [.Mk])(2.16)= ‘Mk‘) A ... A (xd).There are several difficulties in making sense of this expression: first, Mk is notnecessarily a manifold since it may contain reducible connections. Moreover, the formp(xi) A A (xd) may not be compactly supported. Finally, we must guarantee thatthe invariant defined does not depend on the choice of a metric.Chapter 2. Background 16Are there any reducible connections A E Ak satisfying F = 0? For an S’-reductionQ of F, let A be a connection on F induced by a connection on Q. The set of reducibleconnections defined through the reduction Q, is the affine subspace(2.17) {A+iww E f’(X,IR)} C A.As a connection over Q, the curvature F4+i’ = FA + idw represents the first Chern class2irici(Q). As shown by Donaldson [13, Prop. 3] , there is unique choice of a representativeA E AQ for which FA is the harmonic representative of 2?rici(Q). If b(X) = 0, thenthere is a unique ASD connection for each reduction of F. If b(X) 1, then for a genericmetric reducible connections do not appear in the moduli space; that is Mk C B. Ifb(X) 2, then for a generic path of metrics no reducible connections appear in Mk.We use these topological conditions to avoid the appearance of reducible connections inthe moduli space.In general, the moduli space Mk will be non-compact. Uhienbeck [47] shows howa sequence of instantons fails to converge. Each sequence of instantons {[A]} C Mkpossesses a subsequence of representative connections {A } and an ASD connection A,such that for a finite set of points {x1,. . . , x,} the subsequence converges to A in theC°°-topology on X \ {xi,. . . , x}(after appropriate gauge transformations). In the limitthe curvature is(2.18) J = f IIFII2+8K2kX X\{xi Xp} j1for positive integers k, j = 1, 2,...,p. Heuristically, curvature “bubbles off” at the finiteset of points {x1,. . .,x} in the limit. As a corollary of Uhienbeck’s theorem, in [20] anideal compactification IMk of Mk is defined for which the following theorem holds.Theorem 2.1 (Uhienbeck) Let the space IMk be(2.19) IMk = Mk II (Mk_1 x X) II (Mk_2 x s2(x)) ll• II (M0 x sk(x)),Chapter 2. Background 17where si(X) is the set of unordered collections of j points of X. If one gives IMk thetopology so that all sequences converge then Mk C IMk is an open subset with M1,compact.When c2(Pk) = k > (3bt(X) + 5) the lower strata, M3 x s’(X), 0 i < k, havecodimension at least two and Mk carries a fundamental class [Mk] in homology.Donaldson [18, §111] uses the dual point of view to compute the pairing (2.15). Forevery homology class x E H2(X; R) there is a codimension two submanifold V C Mkwhich, when restricted to each strata, is Poincaré dual to the restriction of ,u(x) toH2(M; R), for j = 0, 1,2,. . . , k. Choose the V to be in general position. Assumingthat k (3bj-(X) + 5), the intersection of Vri fl V2 fl. fl V is disjoint from the lowerstrata Mk_3 x s(X), j = 1,2,... ,k, and the set(2.20) VxlflVxfl•••flVxdflMk=VxlflVxl •.f VdflMkis a finite collection of points. Define D to be(2.21) Dx(x1,. ..,xd) = ((x1).. .(xd),[Mk])(2.22)= #(Vr, fl Vx2 fl ... fl V3, fl Mk),where # denotes a count of points with orientation.Donaldson [14, §VI] shows that if b(X) 3, then to every pair of generic metrics g0, g there is a path of generic metrics for which there are no reducible ASDconnections. In this case a cobordism argument shows that the Donaldson polynomialDx : S*(H2(X; Z)) —* Z is independent of the choice of a generic metric.Manifolds of Simple Type and the Donaldson SeriesHere we review the Donaldson series as defined by Kronheimer-Mrowka [39]. The Donaldson series encapsulates the Donaldson polynomials of all SU(2)-bundles over X in aChapter 2. Background 18single function. For manifolds satisfying the so-called simple type condition the Donaldson series has a particularly simple form in terms of the so-called basic classes inH2(X;YZ).Let A(X) be the graded polynomial algebra generated by H2(X; Z) and Ho(X; Z)(2.23) A(X) = S*(H2(X; Z) Ho(X; Z)),where elements in H(X; Z) have degree 4 — i. Let y generate Ho(X; Z) and 1/ = (y) EH4(B*; Z)The Donaldson polynomial is defined on the algebra A(X). If the element Xl .. .A(X) has degree 2a+4b = 2d(k) for some k, the value of the polynomial Dx : A(X) —* Zis given by(2.24) Dx(xi . . . X) = (t(x1) . . . /i(Xa)i/b, [Mkj)Otherwise, Dx(xi . . . Xay1’) = 0.The Donaldson series(2.25) Dx :H2(X;Z)—*Ris defined by Kronheimer-Mrowka [39j to be(2.26) Dx(x) Dx((1 + )expx)where exp x is a formal power series in A(X).Kronheimer-Mrowka [39] define a closed, smooth, simply connected 4-manifold X tohave simple type if the recurrence relation(2.27) Dx(xdLy2)= 4DX(xzy))holds for all x E H2(X; Z) and integers a and b. At this time, it is possible that all4-manifolds may be of simple type. A large class of examples, including elliptic surfacesChapter 2. Background 19and manifolds containing specific embedded surfaces [40], have been shown to have simpletype. The recurrence relations of [22, 21, 39] simplify for manifolds of simple type andshow that the operations of blowing up, blowing down, and rational blowing up anddown, preserve the simple type condition.For a manifold X of simple type Kronheimer-Mrowka [39] prove that there are rationalnumbers m1,. . . , m, R, and cohomology classes K1,. . . , K H2(X; Z) such that forx H2(X; Z) the Donaldson series has the form(2.28) Dx(x) = exp(Ix(x,x))mexp(K(x)),where 1x is the intersection form. The K are integral lifts of w2(X) and are calledbasic classes. These classes express the Donaldson polynomial in terms of topologicalinformation contained in H2(X; Z). The basic classes are conjectured to appear in anexpression from which the Seiberg-Witten invariants [49] are computed.2.1.3 The Chern-Simons Functional cs and Floer Homology HF(Y)We shall now review the development of a cut-and-paste method of computing the Donaldson polynomials. Initially we will look at anti-self-dual connections on the cylinderR x Y for Y a smooth, oriented homology 3-sphere. We use the Chern-Simons functional to construct a complex from which we will compute the Floer homology HF(Y)in the same way that one computes the homology of a finite dimensional manifold fromthe Morse complex. Introduced in 1987 by Floer [29], Floer homology and its role inthe computation of Donaldson polynomials are clearly explained in the excellent surveyarticles Braam [8] and Atiyah [1].Chapter 2. Background 20The Chern-Simons FunctionalHere we review the definition of the Chern-Simons functional. We will need the criticalpoints and gradient flow lines of this functional to compute the Floer homology which isdefined similarly to the homology of a Morse complex.Let Ày be the space of connections on the (trivial) 5’U(2)-bundle Py —p Y and= Aut(Py) the gauge group. For A Ày the Chern-Simons functional is defiled tobe(2.29) cs(A) =_--J tr(dA A A + A A A A A).The differential of the Chern-Simons functional is(2.30) dcs(A) = FA,where the 2-form FA is interpreted as an element of fZ’(Y Adzu(2))* through the pairing(2.31) (, ) =—f tr(w A 7).From this one sees that the critical points of cs are flat connections in A.A smooth path A : [0, 1]— Ày determines a connection on the cylinder [0, 1] x Y. IfWt E Q1(Py, Adzu(2)) determines the vertical projection of the connection A then theform w e ‘(P[o,l]Xy,Adzu(2)) defined as(2.32) w(x,t) = Wt(X)defines a connection A E A[o,lJp. The values of the cs functional at the endpoints of thepath are related by(2.33) cs(Ai)—cs(Ao) =_tr(FA A FA),Chapter 2. Background 21see [8, §2]. If A0 and A1 are gauge equivalent, i.e. the path descends to a ioop in By, theright hand side of (2.33) is the first Pontrjagin class of the bundle over S1 x Y obtainedby gluing by the gauge transformation g e gy such that g A1 = A0. The first Pontrjaginclass of this bundle is equal to 4 times the degree of the gauge transformation g. Hence,the functional(2.34) cs : By —* R/4Zis well-defined.For the functional cs : B —f R/4Z, the critical points are the irreducible fiat connections modulo gauge equivalence. Thus, critical points correspond to irreducible representations of Tri(Y) into SU(2) modulo conjugation by elements of SU(2). Define 7?, Cto be the critical points of es, suggestive of this relationship with the representationvariety. There are perturbations of the Chern-Simons functional making the irreduciblecritical points of es non-degenerate; from now on we assume this to be the case.There is a correspondence between the gradient flow lines of cs on B* and anti-self-dual connections on the cylinder V4RXY modulo gauge equivalence. A gradient flow lineA : R —* .4 satisfies(2.35) = — *3 FAt,with *3 denoting the Hodge-* operator on f*(y). The Hodge-* operator acting on thecurvature of the corresponding connection A€ ARXY is then anti-self-dual,(2.36) =In the reverse correspondence, for an ASD connection A E Ay there is a gauge transformation such that g A is in the temporal gauge. In this gauge, the connection formfor g A has no dt component and g A determines a path A : IR —* Ày. The fact thatA is anti-self-dual implies that A is a gradient flow line.Chapter 2. Background 22The moduli space of ASD connections of bounded energy on the cylinder is equal to(2.37) JJ M(a,,8),a,flE7?.ywhere a and j3 are critical points of cs and(2.38)M(a,) = {A E ARXY P+FA = 0, lim A{t}xy = a, llrnAI{t}Xy= } / c.There is an action of IR on the space M (a, ) by translation so the quotient M(a, i3) =M(a, /3)/R is often used.In defining Floer homology as the homology of the Morse complex for cs there aretwo properties differentiating the construction of a Morse complex for cs : —f IR/4Zfrom the construction for a function on a finite-dimensional manifold.First, cs : —* R/4Z is circle-valued. We will see that this poses no problems if weuse critical points and gradient flow lines in the usual fashion.A second, more serious difficulty occurs in defining an index, 1(a), for a critical pointa. As shown in [8, §3] the Hessian of cs at A is(2.39) : Q’(YAdzu(2)) —* ‘(YAdzu(2)).In the finite-dimensional case the index is defined to be the number of negative eigenvaluesof the Hessian. The operator *dA has a discrete spectrum but has an infinite numberof both positive and negative eigenvalues making the definition of an absolute indeximpossible. Regardless, the definition of a relative index is still possible. Analogous toMorse theory on a finite dimensional manifold we will show how to define a relative index,I(a, ,8), between two critical points a and 3, for which(2.40) I(a,8) = dimM(a,B).Chapter 2. Background 23In [29, §2], the work of Atiyah-Patodi-Singer [2] is used to compute the dimensionof M(a,3). The index is given in terms of the spectral flow SF(cr,3,F) for a pathF: [0, 1] —p Ày with F(0) = c and F(1) = 3. The spectral flow of P counts the numberof eigenvalues passing from positive to negative minus the number passing from negativeto positive as t ranges over [0, 1]. In practise, this definition is modified slightly to dealwith reducible connections, for which *dA has 0-eigenvalues. Let €> 0 be a small positivenumber and define the spectral flow SF(cr, 3, F) for the path F : [0, 1]—* Ày to be thesum of the intersection number of the spectral curves with the line through (0, —E) and(1, e). Let A E M(c, 3) be an ASD connection. In the temporal gauge A defines a pathJ4Figure 2.1: The spectral flow of a path in Ày.A : IR —* Ày. The dimension MA(a, 3) of the component of M(Q, 9) containing A is(2.41) I(a,i3) = dimMA(o,fl)(2.42) = SF(c,i3,A).Given two paths y : [0, 1] —* By connecting a aild 3 E Ry, let -y be the conjunction of the paths 72 -yj’. Lifting the loop F : [0,1] —b By to : [0,11 — Ày, theendpoints of 5’ are equivalent through a gauge transformation g. The spectral flow of Fis equal to 8 deg g. Hence, the spectral flow between two flat connections SF(a, /3), iswell-defined as an element of Z/8Z.Chapter 2. Background 24Floer [29, §2d] shows that the spaces M (of, /3) are canonically oriented, similar toDonaldson’s [17] construction of an orientation for the space of instantons.Floer [29, Prop] shows that M(cr,8) = M(ci,/3)/IR has the structure of a manifold with corners. The boundary ÔM(a, /3) is a union of broken flow lines,(2.43)OM(c,8)= II II M(a,p’) >< r’ x M(pi,p2)x x F xj PI,P2,...,Pjwhere p1,. . . , p3 Ry, and J’!i C & is the isotropy of p.Floer HomologyHere we apply the information on critical points of the Chern-Simons function and itsgradient flow lines to construct a complex similar to the Morse complex.Fixing the trivial connection 0 to have index 0, an index I : —* Z/8Z is definedas(2.44) I(p) = I(p,t9) = SF(p,0).The Floer homology HF(Y) is defined to be the homology of the Z/8Z-graded complex(CF(Y), ãFloer), where(2.45) CF,(Y) = IR(a)yE7Z,I(o)j (mod 8)and the boundary operator, öFloer : CFJ(Y) —* CF3_1(Y), is given by(2.46) aFloer(o) = #i(a,/3)(/3),I(/3)j—1 (mod 8)where # denotes a count with orientation. We sketch an argument of Floer [29, §1]showing ö1Qer = 0. Computing OlOer for a E RF of index j we have(2.47) 0’ioer@) = #M(a,/3).6E7Z,,I3E7Z,I(S)j—2 (mod 8)I(/3)Ej—1 (mod 8)Chapter 2. Background 25For a fixed 6 E 7? of index I(S) = j — 2, the space M(cx, 8) is a 1-dimensional manifoldwith boundary(2.48) II x M(/3,8).i3E7zI(/3)Ej—1 (mod 8)We see that the inner sum of (2.47) is just a count of the endpoints of a 1-dimensionalmanifold with orientation. For each component of M(a, 6), the pair of endpoints haveopposite orientation so the inner sum of (2.47) vanishes for each 8, showing aioer = 0.We will see that Floer homology is the natural space for relative Donaldson polynomialsto take values.Associated to the complex (CF(Y), öFloer) is the dual complex (CF*(Y), dFloer).The cohomology group given by the homology of the dual complex is called the Floercohomology of Y and written HF*(Y). For b,, () HF(Y) and ZSE C (6) EHF*(Y) the natural pairing of HF(Y) and HF*(Y) is given by(2.49) c), b5(6))aE7Z, 6E7Z, oEfl,Changing the orientation of Y changes the sign of the function cs : Ày —* R, reversingthe direction of the gradient flow lines. The index of a flat connection c over Y is(2.50) I(cx) = 5—I(c),and there is a natural isomorphism between the chain complexes (CF*(Y), dFloer) and(CF5_(i), OFloer). The induced isomorphism of the Floer cohomology HF*(Y) and theFloer homology HF5_(Y) is used to define a pairing of relative Donaldson polynomials(2.51) ( , ) : HF(Y) 0 HF5_(Y) —* lItChapter 2. Background 262.1.4 Gluing of Donaldson PolynomialsFor a closed, smooth, oriented 4-manifold X = X1 Uy X2 split by a smoothly embeddedhomology 3-sphere Y with b(X1) > 0 and b(X2) > 0, the Donaldson polynomialDx decomposes into a pairing of relative invariants with values in the Floer homology,HF(Y). We shall continue using the notation of section 2.1.2. Let the orientation of X1induce the orientation of Y by prefixing a frame of TY with an outward pointing normal.All counts in this section are counts with orientation and are denoted with aAs Y is a homology 3-sphere, the vector space H2(X; R) decomposes as(2.52) H2(X; R) H2(Xi; R) H2(X;R).It is sufficient to consider only the case where x1,...,x,. H2(X1)and Xr+1,. . . , X, =Xd E H2(X2) due to the symmetry and multi-linearity of Dx. As we now explain thereare relative polynomials Dx1 : S*(H2(Xi; Z)) —* HF(Y) and Dx2 : S*(H2(X;Z)) —*HF(Y) such that the polynomial decomposes as(2.53) Dx(x1,2.. .,Xd) (Dx1xi,. . .,r),D2(Xr+i,. .,Xd))with (,) the pairing of HF(Y) and HF*(Y).A subspace of the space of instantons may be obtained from gluing instantons from thetwo sides of the homology 3-sphere Y. Define the open manifolds X1 = X1 Uy ([0, cc) xand X1 = X2 Uy ((—cc, 0] x Y) putting a product metric on the cylindrical ends. Foreach p E R define the moduli space of instantons on X1 asymptotic to p by(2.54) M(1,p) {A = 0 and1irnAI{t}y=} / c,and define the moduli space of instantons on X2 asymptotic to p by(2.55) M(2,p) = {A e A2 F = 0 and lirn A{t}xy=} / q.Chapter 2. Background 27The index theorem gives the dimension of the components of M (X1,p) as(2.56) dimM(i,p) —3(1 + 14(Xi))—1(p) mod 8,and the dimension of the components of M((2,p) as(2.57) dimM(X2,p) = —3(1 + bt(X1))—I(p) mod 8.Let M(X1,P)2r denote the component of M(X1,p) of dimension 2r and similarly forM(k2,p)2s. There is a gluing map G defined from(2.58) G: II M(ki,p)2rx M(k2,p)3 Mk.pE7Z,2r+2s=2d(k)which is an injection of the glued connections into the moduli space Mk with open image.Given a sequence of metrics on X which approach the product metric on a long tube[—T, T] x Y between X1 and X2, the image of the gluing map is large enough to containthe intersections of the codimension two submanifolds V C M which are Poincaré dualto the restriction of t(x)€H2(B) to each strata. By abuse of notation we refer to Valso as the codimension two submanifold of M(X1,p) which is Poincaré dual to t(x)when x, E H2(Xi; Z) and similarly for x E H2(X;Z). Note that these submanifolds Vmay be perturbed to be transverse such that(2.59) V n G(M(ki,p) x M(k2,p)) = G(VX fl Mi1,p) x M2,p)),if x, H2(Xi; Z), and similarly for x E H2(X;Z).The relative Donaldson polynomial Dx1 : S*(H2(X; Z)) —* HF(Y) is defined onXi,.. .,Xr EH2(X1;Z)by(2.60) D1(xi,.. . ,Xr) = E #(V1 n V2 n... n Vr fl M(Xi,a)2r)(a),Chapter 2. Background 28where M (X1,a)2T denotes the component of dimension 2r. The map D1 is well definedas a map into HF(Y) by an argument in [15] which is similar to that showing aioer = 0.The polynomial D2 : S*(H2(Xj)) —f HF’) is defined similarly.Now we see how the Donaldson polynomial decomposes along the embedded manifoldY. Let x1,. .. , H2(Xi; Z) and Xrl,. . . , x3 E H2(X;Z); the Donaldson polynomialisDx(x1,... ,Xrs)(2.61) = #(V1 n V2 n . . . V.. n [Mk])(2.62) = n V n . .. Vr+3 fl M(X1,P)2r X M(X2,P)2s)pe1z*Y(2.63) = (Vx, fl . . . V fl M(X, P)2r) #(Vxr+i fl . . . V n M(X, P)2s)pE7Z*(2.64) (Dx1(x1,. . . , xr),D2(r+i,. . . ,2.2 Equivariant Floer TheoryIn this section we describe the extension of Donaidson-Floer theory to one with polynomials taking values in an equivariant version of Floer homology. By extending the ideasof Morse and Floer, Austin-Braam [5] have constructed an equivariant Floer homologyHFg,*(Y) for a smooth, oriented rational homology 3-sphere Y. The gluing maps obtained through HFç,*(Y) enlarge the set of conditions under which we can decomposethe Donaldson polynomial Dx over X X1 Uy X2 to splittings with b(X2)= 0 on oneside or where instantons asymptotic to reducible connections over Y appear. The relativeDonaldson polynomials of Donaldson-Floer theory and the equivariant theory are relatedby an exact triangle.Chapter 2. Background 292.2.1 Equivariant HomologyHere we define equivariant homology and review the Cartan complex through which itis computed. Good references for this section are Atiyah-Bott [6] and Austin-Braam [7].Let G be a compact Lie group and M a manifold with a smooth G-action G x M —* M.The homotopy quotient MG is defined as(2.65) MG = M XG EG,where EG —* BG is the classifying bundle for the group G. The equivariant homologyHG,(M; R) is the homology of the homotopy quotient,(2.66) HG,(M; IR) = H(MG; IR).Some examples demonstrate how equivariant homology sees the structure of both thegroup action and the quotient M/G.If the action on M is free then M x G EG is a fibre bundle over M/G with contractiblefibre EG. Hence M XG EG is homotopy equivalent to M/G and(2.67) HG,(M; R) H(M/G; R).Consider the trivial action of G on a point. In this case {*} XGEG {*} x EG/G =EG/G = BG so(2.68) HG,({*}; R) = H(BG; R),the homology of the classifying space BG of G.The deRham-Cartan complex is a convenient means of computing equivariant cohomology. By representing cohomology classes with differential forms, many cohomologycomputations are expressed in terms of pull-backs and integration. H. Cartan [9] introduced a complex for equivariant cohomology and Austin-Braam [7] a similar complex forequivariant homology.Chapter 2. Background 30The complex for the equivariant homology of a manifold M with a C-action usesequivariant differential forms(2.69) C(M) = Q?(M) = $ (fln-h() 0 ,51i(g))Gj=h+2iwhere g is the Lie algebra of C, S*(g) the polynomials in the dual of the Lie algebra,and n the dimension of M. The superscript C denotes the C-invariant subspace. On aprimitive element w 0 C Q?(M), with w e Qn_h(M) and 4’ C 5(g), the differential 8Gis defined by(2.70) Oo(wO4’)=dwO4’—2KC(wOqS),where C is contraction with the universal element V. The universal element V C 9* ®TMis induced by the infinitesimal action of C on M.2.2.2 Morse-Bott Theory and Equivariant Floer TheoryHere we exploit the structure of By as the quotient space of an SO(3)-action on thespace of framed connections By = A/c to define an equivariant homology theory fory. Just as the Floer homology HF(Y) is an application of Morse theory in an infinitedimensional setting, so too is equivariant Floer homology HFc,*(Y) an application ofMorse-Bott theory in an infinite dimensional setting. The equivariant theory gives gluingformulae for the Donaldson polynomial invariants when reducible flat connections on Ycome into play. A reference for equivariant Floer theory is Austin-Braam [6, 5].Given a function f : M — III, a critical submanifold is called non-degenerate ifthe Hessian restricted to the normal bundle is a non-degenerate bilinear form. If thecritical submanifolds are non-degenerate the homology may be calculated using theMorse-Bott complex. Moreover, there is a complex for computing HG, (M) from a Cinvariant function [6]. We recall here how this applies in the infinite dimensional settingof cs By —* IR/4Z.Chapter 2. Background 31The Chern-Simons functional cs : By —* IR/4Z is invariant under the action of SO(3).After a perturbation of cs the critical points form a discrete set in /SO(3). In thiscase the critical submanifolds in !y are orbits Q SO(3)/P, where P is the isotropygroup of a. For an irreducible connection 0a SO(3). A product connection 0 has{0}.For a homology 3-sphere Y, a complex CFç,*(Y) is constructed using the equivariantdifferential forms on the critical orbits with boundary operator depending on the framedspaces of gradient flow lines, M(a, /3), between critical orbits.The flow lines are given by(2.71)M(a, /3) = {A RxyIF+A = Ay E 0a AI{t}xy e o} 1g.The interaction between the critical submanifolds and the gradient flow lines iscaptured by the endpoint maps u and 1. Define M(a, /3) = M(a, /3)/R. The mapu : M(a,/3) —* °a is defined to be(2.72) u(A) = lim AIft}y,t-*-ooand the map 1: M(a,/3) —* Q is defined as(2.73) 1(A) = lim AI{t}xy.t-*ooDefine a Z/8Z-graded complex by(2.74) CFc,h(Y) =i+jEhmod8with differentialI öSQ(3)(L)®) p=O(2.75) ö(w®q!)=( (_1)(lu*w®q) plChapter 2. Background 32where i’ depends on i and j, andI9so(3) is the boundary operator of equivariant homology.The maps l and u (pull-back and integration over the fibre) act only on the differentialform components of ?(Q,3)•The equivariant Floer homology HFç,*(Y) is defined in [5, §4] to be the homology ofthe complex (CF,(Y),ö) where 9 =The equivariant Floer cohomology HFJ(Y) is defined in a similar way, see [5]. Briefly,the Cartan complex (C(M), ZI) for computing the G equivariant cohomology of a Gmanifold M is defined by(2.76) C(M) 1(M) = (f(M) 0h+2i—jwith coboundary operator ZIG : C(M) —* C1(M). The equivariant Floer cohomologyis the cohomology of the Z/8Z-graded complex(2.77) CF(Y) = c%Q(3)(o)i+jEhmod8I(i3)=jwith coboundary operator ZI=ZI° defined on w ® P 1so(3)(O,3) byI ZIso(3)(W01’) p=O(2.78) ZI°(w ® ‘P) = ( (1)v(uJw 0 P) p> 1In [5, Theorem 2], HF0,(Y) and HF(Y) are shown to be H*(BSO(3))moduleswith a pairing(2.79) (,) : HFq,*(Y) 0 HFJ(Y) —* R.The pairing of a class 0 E Hso(3),(Q), where w E Q*(Q), S*(g), with a class0 HQ(3)( ), where e *(Q) y g*(9*), is given by(2.80) (wOO7) = (,)fA7.We will say more about this pairing in the next section.Chapter 2. Background 33A Spectral Sequence and an Exact Triangle in Floer HomologyThe equivariant Floer homology HFc,*(Y) may be computed through a spectral sequence.We now review the construction of this sequence due to Austin-Braam [5].There is a natural filtration of the complex CFc,*(Y) by the index of the critical orbitsof flat connections.’ The E’ terms of the spectral sequence of this filtration are given bythe equivariant homology of the critical submanifolds [5, Theorem 2], namely(2.81) Ep’q = Hso(3),q(C9a; R).I(a)pmod8The first differential d, is given using 9, = (_1)Ll*u* and is defined on a representative[w 0 ] E Ep’,q by(2.82) di([v]) = [01(v)] [(_1)(l*u*w 0 )] EIn particular, given irreducible connections a and 3 E 7? of index 1(a) = j = I(/3) + 1, ifwe let [vol®1] generate Hso(3),o(Qa;IR) C E and [vol,®1] generate Hso(3),o(Q,;IR) Cthe map(2.83) d,[volc. ® 1] = [(_1)l*U*(VOlcx) 0 1](2.84)= #.iI(,8, a)[vol, 0 1]where # denotes a count of the unframed gradient flow lines with orientation.As an example of computing with this spectral sequence, consider the Brieskornspheres (2, 3, 6k — 1). R. Fintushel and R.J. Stern [26] have enumerated the gaugeequivalence classes of irreducible flat connections over (2, 3, 6k — 1) and calculated theirindices. There are a flat connections at indices 1 and 5, and b flat connections at indices‘The complex is actually constructed on the universal cover of y where the index gives an integergraded filtration.Chapter 2. Background 343 and 7 where±i kodd kodd(2.85) a= and b= 2keven kevenFor each irreducible connection E 7?F we haveR j=O,(2.86) HSO(3),j(Qa) =0 otherwise.The trivial connection hasR j0(mod4),j>0,(2.87) HsQ(3),(O) =—0 otherwise.The E’-term of the spectral sequence for HFç,((2, 3, 6k — 1)) is given in figure 2.2.4R0000000o R R 0 lRb 0 IR 0 Rb0 1 2 3 4 5 6 7Figure 2.2: Spectral sequence for HFq,*(E(2, 3, 6k — 1))We have seen how to compute d1. For [v] e Eq, ifã2(v) = 0, the map ds(v) = [öi(w)1where w E Ep°_4,q+3is such that ôso(a)(w) = ã4(v). As we see in figure 2.2, the generatorsof the standard Floer homology are found in row 0. The component of the map a1 betweenthe equivariant homology of the irreducible orbits Q and Q.. is an oriented count of theChapter 2. Background 35(unframed) flow lines M(p, y) between flat connections p and y, and is identical to thedifferential 0i’ioer for the Floer complex.Now we describe the construction of an exact triangle introduced by Austin-Braam[5], which relates the standard Floer homology HF(Y) and equivariant Floer homologyHFg,*(Y). Notice that for a reducible connection c, the boundary operators ö(Øq) = 0for p> 0, w 0 € sQ(3)(Oa). Hence there is a short exact sequence of complexes(2.88) 0 ISO(3),*(Qa) CFQ,(Y) Q * 0aEflya reduciblewhere Q is the quotient complex. To this short exact sequence there is the correspondinglong exact sequence in homology(2.89)—f e HSQ(3),*(Oa) L HFc,*(Y) - H(Q) -.* HSQ(3)*(Oa) V•aEfly aElZya reducible a reducibleThere is an inclusioll of groups CF(Y) into CFq,*(Y) given by(2.90) n(c) n(vol. 0 1).aE7Z,This inclusion CF(Y) —* CFc,*(Y) composed with the quotient map g is a chain mapbecause the reducible components of 8 map to 0 in the quotient. At the E1-term ofthe spectral sequence, we see that this map induces an isomorphism HF(Y) —* H(Q).When we consider the case where Y is a homology sphere and the product connection isthe unique reducible flat connection modulo gauge equivalence, we get the exact triangleHF(Y)Hso(3),(O) g(1.6) HFç,*(Y).Chapter 2. Background 36We end this section by noting that there is similarly a spectral sequence, (Er, dr), forHF(Y) whose E1-term is given by(2.91) Ei” = H,o(3)(Qa;R).I(a)pmod8There is an adjointness property between the boundary operator of CF,(Y) and thecoboundary operator of CFJ(Y) which continues to hold between the terms of the spectralsequence. In fact, the pairing (, ) : HFç,*(Y) 0 HFJ(Y) —÷ IR is well-defined because,for w 0 q! E CF,(Y) and i 0 E CF’(Y), one has(2.92)Between the terms of the spectral sequence, if [ 0 q] € Eq and [ ® ‘1] Ethen(2.93) (dr([w®q!]),[i®P])It should be noted that Frøyshov [33] has developed a different means of computingthe differentials d’ and d5 in the spectral sequence for HF(Y). This will be useful to usin chapter The Relative Donaldson Polynomial for Equivariant Floer HomologyAssuming b(X1)> 0, a relative Donaldson invariant for a 4-manifold X1 may be definedwith values in HFç,*(Y), where oX1 = Y. For the principal SU(2)-bundle P over= X1 Uy ([0, co) x Y), the SO(3)-space(2.94) M(,a) = {A E = 0, lirnA{}y a mod q} / gChapter 2. Background 37is fibred over both M(f(1,a) and Qa:M(Xi,a)(2.95)where the endpoint map 6c, : M(Z, a) —* Oc is given by e(A) = 1imt AI{t}xy.There is a map /1 : H2(Xi; Z) —* Ho(3)((Xl, a)) defined similarly to the mapH2(Xi; Z) —÷ H2(B1;Z). The space . P Xc Ax1 is the total space of an SO(3)-bundle over Xi x whose projection is equivariant under the SO(3)-action. There isthen an equivariant first Pontrj agin class p’ (E) e H0(3) (X1 xx1). Define(2.96) (x) =pi(i)/x.When bt(X1) > 0, there is a fundamental class -y E Hso(3),*(M(Xl, a)), see [7].The relative Donaldson polynomial Dx1 : S*(H2(Xi; Z)) —* HF,(Y) is defined onxi,...,xrEH2(Xi;7Z)by(2.97) Dx1(x1,. . . , X) = e,((xi) U (x) U U (Xr) fl 7),aEflywhere is integration over the fibre of the projection e : M(X1,a) -.---*Similarly, a polynomial invariant .b2 : S*(H2(X;Z)) —* HF(Y) is defined. Thepairing (2.79) recovers the Donaldson polynomial Dx : S*(H2(X; Z)) —÷ Z from therelative equivariant Donaldson polynomials [5],(2.98) Dx = (.bx1,b2).The exact triangle (1.6) relates the two relative Donaldson polynomials resulting in thepropositionChapter 2. Background 38Proposition 2.2 When bt(X1) > 0, the Donaldson polynomials Dx1 and x1 are related by(2.99) g(bx1)= Dx1.Chapter 3The Chern-Simons FunctionalIn this chapter we show that there is a real-valued lift s of the Chern-Simons functionalcs : By —* IR/4Z for flat connections. The remainder of the chapter develops combinatorial techniques for computing s and applies them to show that there is a uniqueconnectiont€HF1(E(p, q,pqk — 1)) such that s(i) > 0. For us, this is useful as it givesa criterion which tells us that many moduli spaces are empty.3.1 A Real-Valued Lift of the Chern-Simons FunctionalIn this section we construct a lift és of the Chern-Simons functional for the flat connections over the homology 3-sphere Y:R(3.1)By R/4Z.Using the spectral flow of the Hessian of the Chern-Simons functional, we may distinguisha class of paths which we then use to lift cs. The lift recovers an important propertyof the gradient flow of a Morse function on a manifold: the value of the function on agradient flow line decreases in the direction of the flow.To lift the Chern-Simons functional we begin by fixing a trivialization of the principalS’U(2)-bundle Py — Y. This determines a connection 0€ Ày. Choose an elementa e 7? C B. The components of the gauge group are in one-to-one correspondencewith Z through the homomorphism deg : —+ Z. This means that the inverse image of39Chapter 3. The Chern-Simons Functional 40AY/€j.aja /SF=i-8 SF=i SF=i+8 SF=i+16Figure 3.3: The orbit of a connection a E Ày under the action of.a in Ày, which we denote by a, has a countable number of connected components.Let the group ° be the connected component of the identity in . There is one andonly one component °&, with & E a, for which paths F : [0, 1] —* Ày with endpointsA0 g° and A1 = 0 have spectral flow SF(a,0,F) e [0,7J. Choosing such a pathAt: [0, 1] —* Ày, define(3.2) s(a) =— J tr(FA A FA) E R.47r IxYWe will use the lift s because it recovers the monotonicity possessed by gradientflows on a finite-dimensional manifold.Lemma 3.1 Suppose the component of M(a,0) having dimension 0 dimM(a,0) 7is non-empty. Then(3.3) s(a) 0.PROOF: We may calculate our lift using an element A in the component of M(a, 0)having dimension between 0 and 7, as this dimension is the spectral flow of the associatedpath. Because the gradient flow line A corresponds to an ASD connection on R x Y, theChapter 3. The Chern-Simons Functional 41difference in the Chern-Simons functional is given by(3.4)— =— IRxYtr(FA A FA)(3.5)= -±f WFW <0.4IrRXY—Then (a) > = 0.In practise we will use the index calculations for M(c, 8) in the computation of thespectral flow.3.2 Calculations on Brieskorn Homology SpheresThe class of Brieskorn complete intersections and, in particular, Brieskorn homologyspheres is a useful setting for the calculation of the Chern-Simons functional and Floerhomology. The Brieskorn spheres appear in a number of situations— for example, as theboundary of the Gompf nuclei of an elliptic surface, or as the boundary of the canonicalresolution of an algebraic singularity.A class of Brieskorn complete intersections (ai, a2,a3) is the set of 3-manifolds givenby(3.6) {4’ + 42 + 43 = 0} fl 55 c C3,where S5 C C3 is the 5-sphere. If a1, a2, and a3, are relatively prime, then the manifolda2,a3) is a homology sphere.A useful description of the Brieskorn homology spheres is as Seifert fibred spaces. ASeifert fibred space Y is a 3-manifold with a pseudofree circle action—that is, there isan action of S1 on Y whose points with non-trivial isotropy are isolated points whoseisotropy groups are finite subgroups of 5i As a1, a2, and a3 are relatively prime theChapter 3. The Chern-Simons Functional 42circle action on (a1,a2,a3) given by(3.7) (zi,z2,z3)—* (c2a3zi,Aa1a3z,),a1a2z.has three exceptional fibres— each is obtained by setting one of the coordinates z1, z2and z3 equal to zero. As Seifert fibred spaces the Brieskorn homology spheres may beconstructed [46] from the following: their Seifert invariants describing the exceptionalfibres (ai, b1), (a2,b2), (a3,b3); the Euler number, bo; and base space Y/S1.We give (aj, a2,a3) the orientation it receives as the boundary of the canonicalresolution, by prefixing an oriented frame of TE(ai, a2,a3) with an outward normal to(2, 3, 11) to obtain an oriented basis. Orienting (p, q,pqk — 1) in this manner, ourSeifert invariants are (p,p — q*), (q, q — *), (pqk — 1, k) with b0 1, and the base space(p,q,pqk — 1)/S S23.2.1 Flat Connections over E(p,q,pqk —1)We use the correspondence between the flat SU(2)-connections modulo gauge equivalenceover a manifold, Y, and the representations of the fundamental group, p: 7ri(Y) —* SU(2)modulo conjugation. Let r = pqk — 1. When working with a pair of relatively primenumbers p and q, let pp’ 1 mod q with 0 <p* <q and qq* 1 mod p with 0 < q* <p.From the description of E(p, q, r) as a Seifert fibred space we have the followingpresentation of the fundamental group,(3.8)7r1(E(p,q,r)) (x,y,z,hxP = yq q_p*r = hk,xyz = h,h is central).In this section we denote a generator of iri(E(p, q, r)) by lower case letters (x, y, z,and h) and its image under the representation p by upper case letters(X, Y, Z and Hrespectively).Chapter 3. The Chern-Simons Functional 43BTrq17rAfirrCFigure 3.4: Sphere S2 with lines through fixed points of X, Y, and Z.For an irreducible connection, H must be +1 to centralize X, Y, and Z, which arenot contained in the same circle subgroup of SU(2).Consider the following description of the irreducible representations of 7r1(E(p, q, r))into SU(2) given by R. Fintushel and R.J. Stern [26, §2]. Each of X, Y and Z acts onR3 = zu(2) through the adjoint representation. This action induces a covering of SO(3)by SU(2). Associate to each of X, Y, and Z the pair of antipodal fixed points ±A, +B,E 32 of their image in 50(3). These points are distinct because the representationof 7ri((p, q, r)) is irreducible.Through each pair of points in {A, B, C}, draw a great circle. The three resulting great circles divide the sphere S2 into eight triangles. From the presentation ofq, r)) we see that the images of X, Y and Z in S0(3) are rotations of order p, qand r respectively. Let the rotations be through angles of and respectively.The angle between the geodesics A and A at the point A is and similarly at theother intersection points— see Figure 3.4.As the angles of a spherical triangle sum to more than r, for each of the four pairsof antipodal triangles we obtain inequalities involving the rotation numbers 1, m, and n,Chapter 3. The Chern-Simons Functional 44as follows:! + >ir+ir(q-m)+ >(39) p qr(p-l)+ +r(r-.n) >+r(q-m)+These conditions may be rearranged to give the inequalities(3.10) > 1p q r(3.11) < 1p q r1 m n(3.12)———+— < 1p q r(3.13)—++ < 1.p q rGeometrically this constraint on the rotation numbers says that (, !, ) lies withina tetrahedron with vertices at points (1,0,0), (0, 1,0), (0,0, 1), and (1, 1, 1).(1,1,1)(0,0,1) —_________(00/(0,0,0) (1,0,0)Figure 3.5: A tetrahedron in the unit cube.Chapter 3. The Chern-Simons Functional 45Conjugation by elements of SU(2) induce symmetries of the tetrahedron given asrotations by ir about the axes x= y = 1/2, y = z = 1/2 and z = x = 1/2.For given values of 1 and m, the possible values of n/(pqk — 1) which correspond toa points lying in the tetrahedron satisfy(3.14) n pq—mp—lqpqk pq(3.15) n mp+lq—pqpqk pq(3.16) n—i pq+mp—lqpqk pq(3.17) n—i pq—mp+lppqk pqOn the interval [0, 1] the fractions i/(pq) and i/(pq — 1) occur alternately—that isi i i+1 i+1(3.18) —<pq pq—i pq pq—ifor 0 <i < + 1 <pq — 1. Thus the connection nearest the boundary of the tetrahedronin the z-direction will satisfy one of the equations1 m n(3.19)—+—+——— = 1p q pqk(3.20) __:_ = 1p q pqk(3.21) !_+n1 =p q pqk1 m n—i(3.22)——+—+ = 1.p q pqkWhen we talk of a connection near the boundary of this tetrahedron we refer to a solutionof one of the above equations.3.2.2 Index CalculationsFintushel-Stern [26] derive an invariant R(e) of a flat connection over a Brieskorn homology sphere giving the Floer index. The number e is determined from the rotation numbersChapter 3. The Chern-Simons Functional 461 = l, 12 = m, and 13 n, of the flat connection. We orient the Brieskorn homologysphere (ai, a2,a3) by prefixing a frame of TE(ai, a2,a3) with an outward pointing normal of the canonical resolution. For a flat connection 7 E R(ai,a2,3) Fintushel-Stern[26] construct a path {A} in M(7,0) with spectral flow(3.23)1(7) = SF(7,0) = R(e) =+--cot () cot (L) sin2 (lrek)where a =a123and e la/a mod 2a. The Chern-Simons functional on the path{A} is(3.24) cs(-t) = _2fytr(FA AFA)(3.25)aNote that the secondary Seifert invariant b has b:a/aj 1 mod a and also(3.26) be l, mod a.We define the function(3.27) W(a, e) = cot () cot () sin2 ()as a shorthand for the summand of (3.23). Note that W(a, e) depends only on e mod aand the other as’s. The function W measures the contribution to the index from theexceptional fibre of order a of (ai, a2,a3).3.2.3 A Geometric Interpretation of W(a, e)Here we will derive a geometric interpretation for the index in (3.23). The final resultinvolves the difference between the area of a triangle in the plane and the number oflattice points inside it, with a convention for points on the boundary. This formula is theChapter 3. The Chern-Simons Functional 47same as one derived by A.J. Casson and C. Mc. A. Gordon [10] and R. Fintushel andR.J. Stern [25]. Beginning with the equation(3.28) W(a,e) = cot () cot () sin2 (-),we exchange the trigonometric functions for the equivalent expressions involving exponentials. The following substitutions are made: let ,\ = e2Id/a, then(ckir’\ . e1a+ e_I/a(3.29) coti—i = j.a J — 6irick/ae2jck/L + 1 xc + 1(3.30) = .e2DI/cL — 1 — 1and(3.31) sin2(ck/a)= (2i)(eick/a — e)2(62lrick/a— 1)2 — 1)2(3.32) = —_________ ______4e2/a 4)cWe write b* for a/a recalling that for the secondary Seifert invariants ba/a 1 mod a.Chapter 3. The Chern-Simons Functional 48Letting ) range over the a-th roots of unity gives(3.33) W(a, e) =2 :: ‘ . ( 1)2a Z(3 34) — 1b*+ 1 + 1 (,)ebb* — 1)2— 2a )b*— 1 ).— 1(335) = i ( _ebb (b* + 1)(A +1)((be_1)b* ••• + + 1)a((bb*e_1)••• + + 1))(3.36) = (_ 2ebb+ ( (b + 1)( + 1).a\((be_1)b*+••• + X +1)((bb*e_1) + ... + + 1)))(3.37) = (_ 2ebb+ (bbe + 2 _1)b*e + + 2 + 1).a\ Aa=1(_bb*e+ 2A-’ + ... + 2’ + 1))A geometric interpretation of W(a, e) in 3.37 is now derived using the orthogonalitycondition for ,\. For fixed j,I a ifjO (moda),(3.38) = ( 0 otherwise,We shall express the sums in (3.37) as the difference between the number of points in atriangle and the area of the triangle. LetIi ifj0 (moda),(3.39) 6j = ( 0 the 5-function modulo a. Then = a5j.Chapter 3. The Chern-Simons Functional 49Beginning with (3.37) the function W(a, e) becomesW(a,e)2beb* 1 /(3.40) = —_ea+ >2 (bbe + 2_*e •• + 2be + 1)a(A_bb*e + 2A_( *e_i) ••• + 2A1 + 1))2 2beb* 1 / b—i \ / bbe—i(3.41) = e + (2(>2Aib*e) + *e + 1) f 2( >2 A_k) + + ia aAa=i\ j=i / \ k=12 2beb* 2 be—i bbe—i 1 be—i 1 be—i(3.42) = — e +—>2 >2 >2 (b*_k)>2 >2 (j_b)b*e>2 >2 jb*ea a j=i k=1 )a=i a ji a j=i ?=i1 bb*e_i 1 bb*e_i+— >2 >2 A(lb*e_k) + — >2 >2 A_ka k=i A—i a ki A°=i>2 + >2 A_bb*e + >2 1a a a2 2beb* be—i bb*e_i be—i be—i(343) = — e +2>2 >2 6(jb-k) + >2 S(j_b)b*e + >2 S(jb*e) +a j=i k=i j=ibb*e_1 bb*e_i 1 1>2 S(bb*e_k) + >2 6(-k) + (bb*e) + S(_bb*e) + 1.k=i k=iDue to the orthogonality conditions as j ranges from 0 to be and k from 0 to bb*e theonly contributions from the sum occur where jb* — k 0 mod a. Plotting (jb* — k)/aversus j we see in figure 3.6 that all the points of the form (s, t/a) (with s,t integers)inside the parallelogram occur in the sum. In (3.43) the non-zero terms of the first summation correspond to the interior lattice points, while those of the next four summationscorrespond to points on the boundary, and the last 3 terms with the vertices of the parallelogram occurring as lattice points. Symmetry allows us to consider the count on theupper triangle only, being careful to only count one of the lattice points on the x-axis.We omit the point at (0, 0). The function W(a, e) is the same as a count of the latticepoints inside the upper triangle where we weight interior points by 4, points on an edgeby 2, and vertices by 1, minus 4 times the area of the triangle.Chapter 3. The Chern-Simons Functional 50(jb*_k)/a(bebbe/a)(0, _bb*e/a)Figure 3.6: A count of lattice points inside the parallelogram.Chapter 3. The Chern-Simons Functional 51Following Casson-Gordon [10J we define the function 6(x, y) on 1R2 to be the number oflattice points in the right triangle with vertices (0,0), (x, 0), and (x, y) with interior pointsweighted 1, edge point weighted 1/2 and lattice points at the vertices weighted 1/4—and not counting the lattice point (0, 0). The function 5(x, y) makes the relationshipbetween number of lattice points in the triangle and the area of the triangle clearer. Let= 6(x,y)— area(x,y) = 6(x,y)— xy.Our expression for the W(a, e) becomes(3.44) W(a, e) = — (_4b2b*e2 + 8a(be, bb*e/a))(3.45) = 4z(be, bb*e/a).We note that the area of the triangle above isb2*e/a. We rewrite the index in theform:(3.46) R(e)=+ W(a, e)(3.47) = + 4(be,b1’e/a).Some properties of the function (x, y) are useful in our computations. Several ofthese properties can also be derived from (3.23) using the symmetries of cotangent andsine. Expressing the function W(a, e) as a difference between the area of a triangle andthe number of lattice points inside enables further simplification using one’s geometricalintuition. We summarize these in the following propositionProposition 3.2 1. If x andy are integers, then (x,y) = 0.. If x is an integer andy y’ mod x then (x,y)=L(x,y’).PROOF: The first part of proposition 3.2 is a consequence of Pick’s theorem [11]. Weshall compare the count obtained in Pick’s theorem to that for 6(x,y). Pick’s theoremChapter 3. The Chern-Simons Functional 52says that the area of a simple polygon in the plane with vertices on a lattice is equal to1(3.48) area(x,y) = + v—iwhere u the number of lattice points on the boundary and v is the number of interiorlattice points. If x and y are integers, then 3 of the lattice points on the boundary arevertices. Hence there are v interior lattice points, u— 3 lattice points on edges, and 3lattice points at the vertices of the triangle(of which we count 2). The function 6(x, y) isthen(3.49) S(x,y) =(3.50) =(3.51) = + v — 1 = area(x, y)resulting in the first part of proposition 3.2.To prove part 2, it suffices to consider the case when y’ E [0, x) with y y’ mod x.Let n x[] be the largest multiple of x less than y. This lemma amounts to separatingthe count over two pieces — one on the triangle with vertices (0, 0), (x, 0), (x, n) and thesecond over the triangle with vertices (0, 0), (x, n), and (x, y).The count over the lower triangle equals the area by lemma 1. The upper trianglemay be mapped to one with vertices at (0, 0), (x, 0), and (x, y’) by an area preservingtransformation which also preserves the lattice. DIn our calculations on E(p, q,pkq—i), we often consider (ba, bac/p) where ac—np = 1for integers a,b,c,p, and n. Define a new function (x, y) to be the count of lattice pointsinside the triangle of figure 3.8, where we weight interior points 1, edge points , and thevertices each (other than (0, 0)).Chapter 3. The Chern-Simons Functional 53• (x,y)(x.,n)(O,0) • (O,0)_ _.•—j (x, y’)• (x,O) . . •• (x,O)-4Figure 3.7: Reducing (x,y) to /(x,y’).(o,•y) . •:N\ç::(0,0) . • • • (x20)Figure 3.8: The triangle on which (x, y) is defined.Chapter 3. The Chern-Simons Functional 54Lemma 3.3 Let a,c,n and p be integers. ff ba is an integer, then(3.52) L(ba,bac/p) = Z(b,ab/p).We prove this lemma by changing from one basis for the lattice to a more suitablebasis. For the lower triangle of figure 3.9, the value of /(ba, bn) = 0 so /(ba, bac/p) isthe weighted sum of lattice points in the small wedge minus the area of the wedge. Letv = (a,n) and w = (p,c). The map T : R2 — R2 given by(3.53)is area-preserving asT(x,y) = xv+yw(3.54) an = ac — np = 1,and takes the standard integer lattice to itself. The lemma follows as the inverse imageof the triangle ((0,0), (ba, bn), (ba, bac/p)) is ((0,0), (b, 0), (0, ba/p)). DFigure 3.9: A new basis for the integer lattice.(ba,bac/p)(ba,bn)(0, 0)(ba,0)We now calculate L for a particular case to which we shall reduce our calculations.Chapter 3. The Chern-Simons Functional 55Lemma 3.4 For 0 <n <p,(3.55) z(n,n/p) = — —PROOF: The function /(n, n/p) is just a count of points along the x-axis, minus thearea of the triangle. See figure 3.10. D(0’O) (n;n/p)(n,0)Figure 3.10: A count of lattice points along the x-axis.3.2.4 A Distinguished Connection with Index 1Consider the connection t with e = 1. The rotation numbers of the connection areb2 and b3 the secondary Seifert invariants— as seen from (3.26). Using (3.45) andChapter 3. The Chern-Simons Functional 56lemma 3.3 to compute the spectral flow of the path constructed in [26] we find that(3.56) 1(t) = + bb/a)(3.57) = —+4L(1,b/a)(3.58)(359) = 2p_q*q_p* kpq(pqk — 1) p q pqlc — 1(3.60) = 1.The spectral flow is between 0 and 7 so the value of the lift s at t may be calculatedusing the path, and is(3.61)(3.62) = 1/(pq(pqk— 1)) > 0.The remainder of this chapter is devoted to proving the following theorem:Theorem 1.3 There is a unique flat connectioni E HFi((p,q,pqk— 1)) for which thelift is positive.The theorem follows from the next lemma:Lemma 3.5 For all /3 with e as in (3.23) we have(3.63) S(e) = W(p, e) + W(q, e) + W(pqk — 1, e) > 1for 13’.We assume lemma 3.5 for the present and continue with the proof of theorem 1.3.PROOF: Let E = E(p, q,pqk— 1). In [25] to compute the index of /3, Fintushel and Sternconstruct an ASD connection A M(/3,O). The dimension of the component MA(/9,O)Chapter 3. The Chern-Simons Functional 57of M(/3, 0) containing A is given by 3.23. We have(3.64) dim MA(, 0)= pq(p— 1) + W(p, e) + W(q, e) + W(pqk — 1, e)2e(3.65) = + S(e).pq(pqk — 1)Assuming I(3) 1 mod 8, we have2e(3.66) dimMA(/3,0) = 8j + 1> pq(pqk 1) + 1and thuse2(3.67) —4j <0.pq(pqk — 1)Assuming A is in the temporal gauge we have a path of connections A = {A}. Choosea cross-section {t0} x (p,q,pqk— 1).Choose a gauge transformation g of degree —j and a path ‘ : [0, 1] —* A with= A0 and (1) = g A0. This descends to a path y: [0, 1] —* BEConstruct the path A R —f BE to beA tto,(3.68)= y(t — t0) to < t to + 1,t>to+1.The path A is constructed so that SF(A) = SF(A) + SF() = (8j + 1) — 8j = 1.Computing the lift using the path A gives(3.69) s() = Jtr(FAAFA)_4degg(3.70) = —4jpq(pqk — 1)(3.71) < 0,Chapter 3. The Chern-Simons Functional 58by the inequality (3.67). DNow we prove lemma 3.5. We first show the lemma is true for k = 1 and thengeneralize to all k.The case (p,q,pq—1)Recall from our description of the fiat connections on (p, q,pq — 1) that the rotatiollnumbers 1, in, and n are such that (, , -) lie in a tetrahedron in R3. We begin byfirst proving the lemma for connections near the boundary of this tetrahedron.We will use the following number theory exercise presented without proof.Lemma 3.6 If p and q are relatively prime, then(3.72) pp*+qq*=pq+1Recall that(3.73)qq* 1 mod p, and 0 < q* <1 mod q, and 0 <* <qUsing symmetries of the tetrahedron generated by conjugation of representations byelements in SU(2), we need only consider the connections with rotation numbers (1, m, n)such that (.L ) is near to one face: recall from section 3.2.1 that these are givenp’ q ‘ pqk—1by solutions to1 m n(3.74) —+-—+—=1,p q pqor(3.75) ql+pm+n=pq.Chapter 3. The Chern-Simons Functional 59Using the symmetries any connection near the boundary has a representative on thisface.All connections near this face are described by values of e [1, pq — 2], howevernot all of these values of e are realized by a connection. The rotation numbers of theseconnections are 1 _eq* mod p, in ep* mod q, and ii = e, where 0 1 < p and0 <m <q. If + < 1, the connection is a solution of (3.75): working modulo pq wehave(3.76) q(ap— eq*) +p(bq_ep*)+e _e(qq*+pp*) +e (modpq)(3.77) —e(pq + 1) + e 0 (mod pq),and then qi + pm < pq and e < pq imply that ql + pm + e < 2pq so we have a truesolution of (3.75). We shall show that(3.78) S(e) = W(p, e) + W(q, e) + W(pqk — 1, e) > 1for e=2,...,pq—2.Applying lemma 3.3 to the sum S gives(3.79) S(e) = W(p, 3) + W(q, j3) + W(pqk — 1, ,3)= 4(e(p_q*),e(p_q*)(mp_q)/p) +(3.80) 4/(e(q _p*),e(q _p*)(q—p)/q) +4(e,epq/(pq — 1))(3.81) = 41X(e,e(p_q*)/p) +4/(e,e(q_p*)/q) +4t(e,e/(pq_ 1)).Using lemma 2 we replace /(e, e(q — p*)/q) with—(e, e(p*/q)) and get(3.82) S(e) = 4L(e, e(p — q*)/p) — 4L(e, ep*/q) + 4(e, e/(pq — 1)).Observe in figure 3.11 that the triangles in the first two functions are very close: incomputing 8(e, ep*/q) — 6(e, ep*/q) it suffices to count the lattice points in a narrowChapter 3. The Chern-Simons Functional 60triangle. A transformation similar to that in the proof of lemma 3.3 reduces the countto lattice points in the second triangle of figure 3.11.(3.86) = e— q(2q— 1)(3.87) = e (i— q(2q— 1))This last expression (3.87) is concave in e—i.e. the graph of the function lies above anysecant— and is greater than 1 at e = 2 and e = q. Thus S(e) > 1 for e E [2,q].(e2ep*/q)(e, e(p — q*)/p)(0, e,/.p)(0,.0)-4(e/q,0)(°)Figure 3.11: Reduction of a count of lattice points to a small wedge.The sum S(e) becomes(3.83) S(e) _4A(e/q,e/p) +4L(e,e/(pq_ 1)).We separate into two cases for which we apply different estimates. Without loss ofgenerality we assume p < q. First we consider the case p = 2.THE CASE p = 2. For p = 2 we may simply count the lattice points inside each triangle.For e E [2, ql we have(3.84) 8(e) = _4Z(e/q,e/2) +4(e,e/(2q_ 1))/1 e 1 e2’\ / e 1 e2(3.85) >Chapter 3. The Chern-Simons Functional 61A count of lattice points for e [q, 2q — 3] gives (similarly)(3.88) S(e) = _4z(e/q,e/2) +4(e,e/(2q_ 1))(3.89) > -4 ( ( - + (e; 9- G) + (( - - 2(2q- 1))(3.90) = 2q — e— q(2q— 1)The expression (3.90) is concave in e and is greater than 1 for e = q and e = 2q — 3.The value e 2q — 2 does not correspond to an irreducible connection as the rotationnumber 1 e mod 2 0. This establishes the lemma for (2, q, 2q — 1).THE CASE p 3. For e E [2, q] U [pq— q, pq — 2] the proposition can be established bycounting, in a manner similar to the case p = 2. Away from here (when e E [q,pq— qj)we must estimate L more precisely. We shall use the following symmetry property.Proposition 3.7 .(pq — e/q,pq — e/p) = pq/2 — e + (e/q, e/p).PRooF: Considering the figure 3.12, let be the count of points (including the origin) inside the region shown. A count of lattice points for the parallelogram with base(0, 0), (e/q, 0) an height q gives the area. We know the count for the triangle with integervertices gives the area:(3.91) (e/q, e/p) + = eand(3.92) ((pq-e)/q,(pq-e)/p)+E =Eliminate the count and we have the proposition. DChapter 3. The Chern-Simons Functional 62• •‘ pq—e_______• • . . (°‘•°)Figure 3.12: The symmetry in the count 8(e/q, e/p)We return to the calculation of S(e). We see that(3.93) S(e) —47\(e/q, e/p) + 4(e, e/(pq — 1))2e 2e(3.94) =— —46(e/q,e/p) +2e— 1—pq —1(3.95) 2e— 1 — 46(e/q, e/p)_ pq— 1)•By proposition 3.7(3.96) S(pq - e) = 2(pq - e) -1 - 4((pq - e)/q, (pq - e)/p)(3.97) = 2(pq — e) — 1 — (2pq — 4e + 48(e/q, e/p))— pq(pq —1)(3.98) = 2e -1- 46(e/q, e/p)-1)+ 42pq(3.99) = S(e) — 2pq — 2epq —1For the values of e [q,pq — q], <2. We have shownProposition 3.8 For 1 e pq/2,(3.100) S(pq — e) = S(e) —Chapter 3. The Chern-Simons Functional 63Our strategy is to prove that S(e) > 3 for e e [q,pq/2]. We then use the symmetryin Proposition 3.8 to conclude that S(e) > 1 for e [pq/2,pq— q].Proposition 3.9 For p < q, we have (e/q, e/p) (e/p)(e/q + 1) + .PROOF: We compare the count 6(e/q, e/p) to the area of the trapezoid with verticesat (0, 0), (e/q + 1/2, 0),(1/2, e/p) and (0, e/p), as in figure 3.13. The area of the rectangle about each lattice point equals its weight. On each vertical lattice edge, x =the weighted sum of the lattice points is bounded by the area under the trapezoid andbetween x = n — and x = n + . The exceptions are near the y-axis and near (e/p, 0).Near the y-axis, the lattice point (0, 0) is not included in the count so the area under thetrapezoid with 0 < x < still bounds the weighted sum along the y-axis. The piece ofthe box around ([], 0) may contribute up to an extra beyond the area of the trapezoidaccounting for the extra in the statement of the proposition. 02e(°‘.°)12xFigure 3.13: Estimation of 6(e/q, e/p)Chapter 3. The Chern-Simons Functional64We are ready to estimate S(e). We have(3.101) S(e) — 3 = —4(e/q, e/p) + 4(e, e/(pq — 1)) — 3(3.102) > -4 (ePei+1) +-(3.103) = —5+ (2—e—2e2.p (pq—l)The estimate on S(e) is concave, so we needonly check e = p and e = pq/2. When e =we have(3.104) S(e) —3 > 0 —5+ 2q (i——1> 0(3.105) —5p(pq — 1) +2q(pq—l)(p — 1) — 2pq >0(3.106) 2(pq — 6)(q —5/2)p + (lOpq + lOq — 35) + 2q> 0.The last expression is clearly true for p 3 and q 4. A similarexpression may bederived for e = pq/2. An application of proposition 3.8 shows that the lemma holds fore e [2,pq —2].Having shown that the lemma holds for flat connections near the boundary of ourtetrahedron we may conclude it is true for all flat connections on E(p, q, pq —1) by utingthe concavity of W(pq — 1, e) with respect to rotation number n e mod pq—1. Wehave(3.107) 4(n,n/(pq - 1)) = 4((n -1)+ - 2(- 1)(3.108) = 2n—1—’1which is concave. Holding the value of the rotation numbers 1 and m constart weconsider e as function of n. If thesum S(e(n)) > 1 for the extremal n, thenS(e())> 1for all n realized by a connection.Chapter 3. The Chern-Simons Functional65One exceptional case remains, the connection3 near t with rotation numbers 1 =— m q — p, n 2 and corresponding to e = pq +1. We must show separatelythat 3 has S(e) > 1 as the concavity argument does not work for the connection nearestc. When we compute /3 from (3.83) and (3.108) we get(3.109) S(pq+l) = W(p,l) +W(q,l)+W(pq— 1,2)(3.110) = --+3—8pq pq—lwhich is greater than 1 when pq 6.The general case: YZ(p,q,pqk —1)We are ready to prove the lemma for (p, q, pqk — 1).We use the following lemma to compare S(p,q,pq.1)(e)for the connection with rotationnumbers l,m, and n over (p, q, pq — 1) to SE(p,q,pqk_1)(e)for the connection with rotationnumbers l,m, and kri over E(p,q,pqk — 1).Lemma 3.1.0 z(kn, knpq/(pqk — 1)) > /.(n, npq/(pq—1)).PROOF: Applying lemma 3.3 to both expressions, we have(3.111) L(kn,knpq/(pqk— 1)) = z.(n,nk/(pqk— 1))and(3.112) /..(n,npq/(pq —1)) = (n,n/(pq —1)).As the rotation number n is between 0 and pq—1 by lemma 3.4 we haven 1(3.113) z(n,npq/(pq—1))n 1 kn2(3.114) 2 4 — 2(pqk — 1)(3.115) = /(kn, knpq/(pqk —(3.116)Chapter 3. The Chern-Simons Functional 66We conclude from lemma 3.10 that lemma 3.5 holds for connections on (p, q, pqk —1)with rotation numbers i with 1 0 mod k. Using a symmetry of the tetrahedron weconclude the result holds for connections on E(p, q, pqk — 1) with rotation numbers ijwith i k — 1 mod k.We now use concavity again. For a rotation number n e [sk, (s + 1)k—1], we have(3.117)(n, npq/(pqk — 1)) = (sk, skpq/(pqk — 1)) + (n — sk) — 1 + — ( sk2which is again concave as a function of n. Holding the value of 1 and m fixed consider eas varying with n. For both n = sk and n sk + k — 1, the function S(e(n)) > 1 and bythe concavity of S(e(n)) we may conclude that S(e(n))> 1 for all n E [sk,sk + k — 1].Again we must show the lemma for a single exception: the connection /3 near t withrotation numbers 1=p — q* m = q — p, = k + 1 and corresponding to e = pqk + 1.The connection 3 must be shown to have 5(e) > 1 as the concavity argument does notwork for the connection nearest t. When we compute j3 from the analogues of (3.83) and(3.108) we get(3.118) S(e) = +2k + 7 — 2pq(k +1)2>pq pqk—1aspq6. D3.2.5 Monotonicity of the Chern-Simons FunctionalHere we derive a corollary of theorem 1.3 which will be useful in the next section. Thecorollary demonstrates the power of our construction of the lift of cs in determining themoduli spaces of ASD connections on the cylinder Z x E(p, q, pqk — 1) to be empty.Corollary 1.4 There is only one nonempty one-dimensional moduli space M(o, ) ofinstantons on the cylinder R x (p,q,pqk — 1).Chapter 3. The Chern-Simons Functional 67PROOF: The statement of the theorem may be reinterpreted as the following: The spaceM(i, 0) is the single non-empty space of ASD connections between flat connections ofindex 1 and the product connection 0. This follows immediately from lemma 3.1 andtheorem 1.3 as t. is the only flat connection of index 1 with cs(t) > 0. DChapter 4Computing Donaldson Polynomials using Equivariant Floer TheoryIs the Floer homology HF(Y) the span of the elements Dw where W is an oriented, simply connected 4-manifold with oW = Y? In this chapter we show the answer is “no”: forthe Brieskorn spheres (2, 3,6k—i) there are elements ofHF4÷1(E(2, 3,6k—i)) which arenot realized by Donaldson polynomials. Specializing to the Brieskorn sphere (2, 3, 11),we show that the component of the Donaldson polynomials in HF4+1((2,3, ii)) is zero.For the Brieskorn homology spheres E(2, 3, 6k — 1), P. Lisca [42] has shown a complementary result. Recall that HF(E(2, 3, 6k — i)) = 0 if i is even. Lisca shows thatHF4+3((2,3, 6k — 1)) is generated by the image of the Donaldson polynomials of theGompf nuclei.Recall from our discussion in section 2.2.2 that we may define the relative Donaldsonpolynomial for the equivariant Floer groups as well. The polynomial Dx factors in thefollowing way,3 DxH0(3),* (0)HFc,*(Y) 3 Dx(4.i)where g o Dx = Dx. The space Hso(3),(0) R[u], where u has degree 4, is theSO(3)-equivariant homology of the trivial connection, 0. Define the subspace V C68Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 69HF(Y(2, 3,6k — 1)) as in the introduction, where(4.2) V = Image(g),and recall from the exact triangle (1.6) and proposition 2.2 that V contains the image ofthe Donaldson polynomials. For the Brieskorn homology spheres = (2, 3, 6k — 1) weexhibit [t] HF1(), [v] E HF5(E) of indices 1 and 5 which are linearly independent inHF((2, 3,6k— 1))/V. The elements t and v are thus not in the image of the Donaldsonpolynomial for any 4-manifold X.Specializing to the case (2, 3, 11) whereIR ifi=l,3,5,7(4.3) HF((2, 3, 11)) =0 otherwise,will give us corollary 1.2 from theorem 1.1.The element i. is the unique fiat connection of index 1 for which the lift of the ChernSimons functional is positive—a fact essential to the proof that t. g(HFq,1(E)). Theclass v is a formal sum of fiat connections of index 5, which appears as the image of a Donaldson polynomial for the canonical resolution of the (2, 3,6k — 1)-singularity. To showv ‘ V, we can show that 6(i) 0 and 6(v) 0, or equivalently we may explicitly compute the image of the volume forms [vol, 01] E Hgo(3),1(O,) and [vol 01] E Hso(s),5(Ounder the maps d1 and d5 (respectively) in the spectral sequence for HFc,*(). Here wechoose the latter point of view as an example of calculating with the Cartan complex forHFg,* (s).4.1 The Canonical Resolution of the (2,3,6k — 1)-singularityHere we present a few facts about the canonical resolution, C, of the (2, 3, 6k — 1)-singularity.Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 70.ooooooo(k — 2) (—2)-spheres—2Figure 4.14: The surgery diagram for the canonical resolution of (2,3, 6k — 1).Recall that the Brieskorn sphere (2, 3, 6k — 1) was defined as a subset of C3 by(4.4) {(z1,z2,z3)z + z + z’ = o} n S5For the algebraic variety(45) Vk = {(zi,z2,3)Iz+ z23 + z1 = o}there is a canonical resolution of the singularity at 0, r : Vk —* Vk. Let the 4-manifoldC = 7r’(Vk fl B6), where B6 is the unit ball in C3, and note that 9C = (2,3,6k— 1).Moreover, the intersection form Ic is negative definite. By abuse of terminology wewill sometimes refer to the manifold C as the canonical resolution of the singularity(2, 3, 6k — 1). Following [45] we may calculate a surgery description of C. For k 2 thesurgery description is given in figure 4.14.Each —2-sphere in the surgery picture corresponds to a class e E H2(C; Z) with e• e =—2. From figure 4.14, it is seen that the intersection form Ic : H2(X; Z) x H2(X; Z) — Zhas the form,Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 71The intersection form has the following decomposition,Proposition 4.1 The intersection form Ici is equivalent to(4.7) (k — 1)(—1).PROOF: The lower right 8 x 8 submatrix of Ic is the intersection matrix for theE8-form.The following change of basis matrix accomplishes the reduction of Ic to the requiredform. The ninth column from the right is chosen to be orthogonal to the lower 8 rows of(4.6)/Ic=—2 1 0 ... 0 0 0 0 0 0 0 0 0 01 —2 1 ... 0 0 0 0 0 0 0 0 0 00 1 —2 : : : : : : : :.. 10000000000 0 ... 1 —2 1 0 0 0 0 0 0 0 00 0 ... 0 1 —3 1 0 0 0 0 0 0 00 0 ... 0 0 1 —2 1 0 0 0 0 0 00 0 ... 0 0 0 1 —2 1 0 0 0 0 00 0 ... 0 0 0 0 1 —2 1 0 0 0 00 0 ... 0 0 0 0 0 1 —2 1 0 0 00 0 ... 0 0 0 0 0 0 1 —2 1 0 10 0 ... 0 0 0 0 0 0 0 1 —2 1 00 0 ... 0 0 0 0 0 0 0 0 1 —2 00 0 ... 0 0 0 0 0 0 0 1 0 0 —2Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 72the matrix for 1c of (4.6), and splits off theE8-form:4.2 Superfluous Elements of Floer HomologyIn this section we prove the following proposition,Proposition 4.2 There are elements t E HF1(), v€HF5(E) such that in the exacttriangle (1.6) we have1. 6(i)02. 6(v)#0.‘I(4.8) A=100 ...000000000011 0...0000000000011...000000000000... 1 100000000000... 0 110000000000... 0021000000000... 0030100000000... 0040010000000... 0050001000000... 0060000100000... 0040000010000... 00200000010\00... 00300000001IIn particular i. g(HFç,i(E)) and v g(HFç,s()).Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 73This proposition in then used to prove theorem 1.1 by showing the span of the Donaldsonpolynomials lies in a proper subspace of HF(E(2, 3, 6k — 1).These arguments are very much in the spirit of those of Fintushel-Stern [23, 24].In each case a specific moduli space is used to construct cobordisms which allow us tocompute the boundary operator a of the equivariant Floer homology. We compute 8explicitly for tand use an invariant due to Frøyshov [33] to compute 8(v).The E’-term of the spectral sequence for HFç,((2, 3,6k — 1)) is given in figure 4.15,where!±i kodd kodd(4.9) a= 2 and b= 2keven kevenas computed in [25].o R1a Rb 0 R 0 Rb0 1 2 3 4 5 6 7Figure 4.15: The spectral sequence Eq for HFç,*(E).We are illterested in the map d1 : —* E,0. We recall from section 2.2.1 that(4.10) d1([vol 0 1]) (_1)v[lu*(vol) 0 1]where u : M(t, 0) — 0, SO(3), and 1: M(’ 0) —* {0}. The map d1 counts thecomponents of M(i, 0), or equivalently the points of M(t, 0), with orientation.Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 744.2.1 Computing d1Here we prove the first part of proposition 4.2; that 0(t) 1. The proof uses ideassimilar to Fintushel and Stern [24] where reducible connections appear as endpoints of a1-dimensional moduli space.Choose e E H2(C; R) such that e e = —2, where e is a class in the subspace ofH2(C; Z) supporting theE8-form. Let L — C be the line bundle with ci(L) e, E bethe SO(3)-vector bundle L IR and let Q be the associated principal SO(3)-bundle toE. The dimension of the moduli space(4.11) M(C,O) = {A E AQIF = O,A80 =is given by (2.14)(4.12) dimM(C,O) = —2p1(E)— 3(1 + b1 + b) = —2e2 .3 = 1.Once a generic metric is chosen the components of M(C, 8) consist of circles, and open,closed, or half-open arcs. The closed ends of M(C, 0) correspond to reducible instantons.The open ends of M (C, 8) correspond to factorizations through a flat connection a E l?Eof index 1,or elements of M(C, a) x M(a, 0). The following lemma will tell us how manyreducible instantons there are.Lemma 4.3 Let E L IR over C, where L is a line bundle with e = ci(L) andIc(e, e) = —2. If for a line bundle L’ we have E L’ R then(4.13) ci(L) =or in other words, the reduction is unique.PRooF: Let f = ci(L’). From the classification of SO(3)-bundles over C (refer toDold-Whitney [12] and Fintushel-Stern [24]) we have(4.14) e f mod 2Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 75and(4.15) Ic(e,e) Ic(f,f).From (4.14), assume that f = e + 2h where 0 h e H2(C;Z). Equation (4.15) nowreads(4.16) Ic(e,e) = Ic(f,f)(4.17) = Ic(e, e) + 4Ic(e, h) + 410(h, h)or(4.18)— Ic(h, h) I(e, h).Consider —I as a positive definite bilinear form on H2(C; R) with norm lxii =—Ic(x, x) for x e H2(C; R). Using (4.18) and the Schwartz inequality one gets(4.19) 11h112 = —Ic(h, h) ilc(e, h)I = — e• hi lieu ihjIor(4.20) ihjI <On a real-vector space endowed with a definite bilinear form, if we have equality ihil =Well in the Schwartz inequality then h = )e for ). E IR. Now (4.18) implies that = —1,or stated another way, that f = —e.If 11h112 < hell2 = 2, then Ic(h, h) = —1 and because(4.21) Ic (k — 1)(—1) e E8,by proposition 4.1, the class e being in the support of theE8-form implies that Ic(e, h) =0 for all classes h of self-intersection I(h, h) = —1. Now (4.18) implies Ic(h, h) = 0, aChapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 76contradiction. DLet us return to our study of the moduli space M(C, 0). The single reducible occurringin M(C, 0) is the reducible ASD connection u associated to L. The open ends of M(C, 0)correspond to connections bubbling off at öC = . We have(4.22) OM(C, 0) = II fi M(c) x M(cr, 0)I(a)=1where M(a) denotes the 0-dimensional component. As shown in corollary 1.4, = t isthe only flat connection of index 1 for which M(a, 9) 0.As a 1-dimensional manifold such as M (C, 9) must have 0 ends counted with orientation, the cardinality #M(C, t). #M(i, 0) = ±1. This says that #M(t, 9) = ±1 showingthat di([vol10 1]) = +1, or 0, and implying that ã(i) Frøyshov’s Invariant and the Elements of HF5((2, 3, 6k — 1))Here we state some results of Frøyshov [33] on computing equivariant Floer cohomology. First, a map 4 determining the contribution of a product connection 9 E ‘l?y tothe equivariant cohomology is introduced. Second, in [33] the value of this map on anappropriate 0-th degree relative Donaldson polynomial is calculated. After computingthese maps for the manifold C, we use the duality between equivariant Floer homologyand equivariant Floer cohomology to show that the image of the appropriate 0-th degreeDonaldson polynomial under d5 is non-zero.Let Y be a smooth, oriented homology 3-sphere. To compute equivariant Floer cohomology Frøyshov adjoins to the Floer cohomology HF*(Y) a contribution from theproduct connection 0. For the connection 0€‘R.y with index 1(0) = 0 he defines a mapChapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 774, for 0 <j <4, taking the equivariant cohomology of the orbit Qe to HFJ+1(Y),(4.23) 4 : HQ(3)(Oa) HF’(Y).In [33, Theorem 1.1, p. 18], the inclusion of HQ(3)( c) into E10,* in theE1-term of thespectral sequence for HFJ(Y) results in the following commutative diagram relating 4and d, the differential of the Er-term of the spectral sequence for HF(Y): for r 2,H%)(Oe) HF(Y)(4.24)E°”—1 -* E°.Frøyshov computes the value of the map 4 on a 0tldegree Donaldson polynomial.Let X1 be a smooth, oriented, simply connected, negative-definite manifold X1 withboundary Y, where Y is a homology 3-sphere and define X1 = X Uy ([0, ) x Y). Principal U(2)-bundles U over are classified by their first Chern class ci(U)€H2(X; Z).Because Y is a homology 3-sphere, for a U(2)-bundle U there is a reduction of the bundleover the cylindrical end to the principal SO(3)-bundle over [0, ) x Y. Similar to themoduli space of an SU(2)-bundle over X1, there is a moduli space of instantons over U.As S1 x SU(2) doubly covers U(2), the model for the affine space A1(u) decomposes as‘(X1,u(2)) = Q1(X,iR)V(X2,su(2)). Let M(X1,ci(U), c) be the set of connectionsA E Au with a fixed central part and such that P+FA = 0, modulo gauge equivalence.A relative Donaldson invariant D1, (U) : A(X1) —* HF (Y) is defined similarly to thatfor SU(2)-bundles in section 2.1.4. Using a cobordism argument between the reducibleconnections in the moduli space, Frøyshov computes the pairing between 4(v), for agenerator v H.Q(3)(Qs), and Dx1,2(u). The value of this map is a topological invariant of X1 and depends on the number of S’-reductions of U. For w E H2(X; Z) anda E Ry define(4.25) R(X,w,O) ={{z1,z2}cH2(Xi;Z)Izi+z=w,zi.z0}.Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 78In [33, Theorem 3.1, p.56], it is shown that for the generator u E H4(BSO(3); Z) =H,0(3)(O) the value(4.26) (Dx1,,(u)(l), d(u)) = #(RCX1,ci(U), 0)).We now specialize to the manifold C with boundary E(2, 3, 11) to show part 2 ofproposition 4.2 by computing the value of the map d5 : E0 —* E,4 on an element[v] e E,0 in the spectral sequence. First we count the elements of R(C, c, 0) for aspecific w E H2(C; Z).Proposition 4.4 Let C be the canonical resolution of the (2, 3, 6k — 1)-singularity. Ifw = 2e H2(C;Z) where Ic(e,e) = —1 then(4.27) #(R(C, w, 0)) = k — 1.PROOF: In the case = C and w = 2e E H2(C;Z) recall that 1(0) = 0. We have(4.28) R(C, w, 0) = {{z1,z2} zi + z2 = w, Ic(zj, z2) = o}.Let z1 = ne + x where Ic(e, x) = 0. Then z2 = —(n — 2)e — x to satisfy the requirementz1 + z2 w = 2e. The intersection number is(4.29) Ic(zi,z2) Ic(ne+x,—(n—2)e—x)(4.30) = —n(n—2)Ic(e, e) — (2n — 2)Ic(e, x) — Ic(x, x)(4.31) = n(n — 2)— Ic(x, x).As Ic is negative definite we there only two possibilities to obtain I(zi, z2) = 0:• n = 1. Hence Ic(x, x) = —1 and one gets elements of the form(4.32) {e+x,e—x} where I(x,x)—1There is one pair for each diagonal term (—1) in the intersection form. Fromproposition 4.1, we conclude there are k — 2 such classes orthogonal to e.Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 79• n = 0 or 2. In this case Ic(x, x) = 0 which implies x = 0. The only pair is {2e, 0}.We have that #(R(C,w,O)) = k—i.Let v = Do,,L, (1). The class w is chosen so that the 0-th degree Donaldson polynomialhas values in HF(Y) at index I such that the(4.33) 0 —2w — 3(1 + b(C)) — I mod 8or as C is negative definite, I 5 mod 8. By [33, Theorem 3.1, p.56] we have(4.34) k —1 = #(R(C,w,O)) = (v,4(u))I(4.35) = (v,d5i.t))I= Kd5(v),u>I,where we have used the fact that the 0 of CFç,(Y) and the d of CFQ(Y) are adjointswith respect to the pairing of HFc,*(Y) and HF(Y), see section 2.2.2 and [5]. HenceId5(v) k — 1 times a generator of Hso(3),4(O) and is non-zero, completing the proof ofproposition 4.2. D4.2.3 The Image of the Donaldson PolynomialsWe now prove theorem 1.1 from the previous proposition.Theorem 1.1 There is a subspace V C HF4+1(E(2, 3, 6k — 1)) of at least co-dimensiontwo such that the following holds. If X1 is a compact, simply-connected, smooth, oriented4-manifold with OX1 = (2, 3,6k — 1), then the component of the Donaldson polynomialD, : S*(H2(Xi;Z)) —+ HF41((2,3, 6k — 1)) lies in V. That is,S*(H2(Xi;Z))—* V C HF4+1(E(2, 3, 6k — 1)).Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 80PROOF: Define the subspace V of HF(Z(2,3,6k — 1)) to be(4.36) V = Image(g).Recall proposition 2.2 on the factoring of the Donaldson polynomial through the exacttriangle,HF(Y)Hso(3),(0) g(1.6) HF,(Y).Hence, the Donaldson polynomial Dx1 for a 4-manifold X1 with oX1 = (2, 3, 6k — 1)factors as(4.37) S*(H2(X1,Z))‘ HFc,*(E(2, 3, 6k —1)) V C HF((2,3,6k — 1)).To prove the theorem we need only show the inclusion V C HF((2, 3, 6k — 1)) is strict.Proposition 4.2 gives us elementst HF1(>(2,3,6k —1)) and v HF5((2,3,6k —1))with 8(t) 0 and 6(v) 0. Exactness of the triangle (1.6) implies t V and v V. C4.3 Applications to (2,3,11)Here we apply the proposition 4.2 to the Brieskorn sphere (2, 3, 11). We obtain information on the Donaldson series of a smooth 4-manifold X with a smoothly embeddedE(2, 3, 11) such that X = X1 UE(2,3,11) X2.First we show a corollary of theorem 1.1 in the special case k = 2.Corollary 1.2 If X1 is a compact, simply-connected, smooth, oriented i-manifold withOX1 = (2, 3, 11), the component of the Donaldson polynomial Dx1 : S*(H2(Xi; Z))HF4+(>Z(2,3,6k — 1)) is zero.Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 81PROOF: Recall from [26] thatR if i = 1,3,5,7(4.38) CF(E; IR) =0 otherwise.with OFlOer 0. The connection t generates CF1(; IR) and v E HF5(;IR) is a multiple ofthe generator p CF5(Z; IR). From theorem 1.1, the image of Dx1 is strictly contained inHF4+1((2,3, 11)). Since this space is 2-dimensional, the only possibility is V = (0). 0We note that since the group Hso(3),(O) is supported only in the dimensions congruentto 4, through the short exact sequence(4.39) 0 = Hso(3),l(O) L HFg,i - V = 0we may conclude that HFc,4*+i(E(2,3, 11)) = 0.Because the relative Donaldson polynomial D, of a compact, oriented 4-manifoldX1 with boundary (2, 3, 11) is only supported in HF4+3((2,3, 11)) this restricts thenumber of classes x1,. . . , x, E H2(X1) for which D1(x1,. . . , x,) 0. To illustratethis limitation, consider the embedding of (2, 3, 11) in the elliptic surface E(2) withb(X2) 2 mod 4. The surface E(2) M(2, 3, 11) UE(2,3,11) N where N is the Gompfnucleus, and the space M(2, 3, 11) is the Milnor fibre of the singularity[36]. We have14(M(2, 3, 11)) = 2 and b(N) = 1. As b(X1) 2 mod 4 for a connection a Ethe dimension of the moduli space M(M(2, 3, 11), a) is(4.40) dim M(M(2, 3,11), a) —3(1 + b(M(2, 3,11))) — 1(a) mod 8.By corollary 1.2 the contributions to the relative Donaldson polynomial DM(2,3,11) maycome only from the connections a of index 1(a) 3 mod 4. In this case the dimensionof M(M(2, 3, 11), a) is congruent to 0 modulo 4. In order for the pairing(4.41) (IL(xi) A . A u(x), [A’t(M(2, 3, 11), a)])Chapter 4. Computing Donaldson Polynomials using Equivariant Floer Theory 82to be a finite set of points, the number of classes n must be even. Similar restrictions canbe obtained for every smooth 4-manifold bounded by E(2, 3, 11) but we have been unableto construct a stronger vanishing condition. We suspect that a vanishing theorem willresult if either relative Donaldson polynomial Dx1 has support only on an odd numberof classes x1, . . . , x, where n 1 mod 2.Bibliography[1] M.F. 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