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APPEARED IN BULLETIN OF THE

arXiv:math/9204226v1 [math.DG] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 317-321THE ATIYAH-JONES CONJECTUREC. P. Boyer, J. C. Hurtubise, B. M. Mann, and R. J. MilgramAbstract.

The purpose of this note is to announce our proof of the Atiyah-Jonesconjecture concerning the topology of the moduli spaces of based SU(2)-instantonsover S4. Full details and proofs appear in our paper [BHMM1].1.

IntroductionIn 1978 Atiyah and Jones [AJ] proved a foundational theorem on the low-dimensional homology of the moduli space of based SU(2) instantons on S4 andmade a number of conjectures. In this note we briefly explain how, using methods ofalgebraic geometry and algebraic topology, we analyze the global geometry of thesemoduli spaces, which we denote by Mk, and prove the Atiyah-Jones conjectures.More precisely, let Pk be a principal SU(2) bundle over S4 with second Chern classk.

Let Mk be the space of based gauge equivalence classes of connections on Pkthat satisfy the Yang-Mills self-duality equations of SU(2) gauge theory. There isa natural forgetful map ϑk : Mk →Bk where Bk is the space of based equivalenceclasses of all connections in Pk.

Atiyah and Jones [AJ] showed that Bk is homotopyequivalent to Ω3kSU(2) and provedTheorem [Atiyah-Jones]. ϑk induces a homology surjectionHt(Mk)(ϑk)t→Ht(Ω3kSU(2)) →0for t ≤q = q(k) ≪k.The main point of the Atiyah-Jones result is that the homology of Ω3SU(2)is completely known so that the surjection sheds some light, for small values oft relative to k, on the heretofore unknown groups Ht(Mk).Notice that, as kincreases, the target spaces Ω3kSU(2) ∼= Ω3k+1SU(2) remain homotopy equivalent,whereas the topology of the Mk, which are 4k dimensional complex manifolds,changes as k varies.

Atiyah and Jones then made the following conjectures:1. (ϑk)t is a homology isomorphism for t ≤q(k).2.

The range of the surjection (isomorphism) q = q(k) can be explicitly deter-mined as a function of k.3. The homology statements can be replaced by homotopy statements in bothconjectures 1 and 2.1991 Mathematics Subject Classification.

Primary 32G13, 55P35; Secondary 14F05, 81E10.During the preparation of this work the first and third authors were supported by NSF grants,the second author by an NSERC grant, and the fourth author by MSRI and an NSF grantReceived by the editors August 14, 1991c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2C. P. BOYER, J. C. HURTUBISE, B. M. MANN, AND R. J. MILGRAMThe last and strongest statement became commonly known as the Atiyah-Jonesconjecture.Notice that, while it is a simple fact that there are natural mapsjk : Ω3kSU(2)≃→Ω3k+1SU(2), there is no obvious map on the Mk level that raisesthe instanton number k. Rather, it follows from analytic results of Taubes [T1]that there are compatible inclusions ιk : Mk →Mk+1 on the instanton modulispace level such that the following diagram homotopy commutes(1.1)Mkιk→Mk+1yϑkyϑk+1Ω3kSU(2)≃→Ω3k+1SU(2)This diagram permits one to take direct limits and hence formulate the stableversion of the Atiyah-Jones conjecture.

This was proven by Taubes [T2].Theorem [Taubes Stability Theorem]. Let M∞be the direct limit of the Mk’sunder the inclusions ιk and let ϑ : M∞→Ω30SU(2) be the direct limit of the mapsϑk in diagram (1.1).

Then ϑ is a homotopy equivalence.Using different methods Gravesen [Gra] independently proved that ϑ is a homol-ogy equivalence. However, these are stable results and do not directly address thestructure of Ht(Mk) or πt(Mk) for any fixed t or k. In another direction, [BoMa]showed that (ϑk)t is a surjection in Z/p homology for t ≤k but the techniques usedthere do not extend to analyze the kernel.In [BHMM1] we prove the following theorem, which settles conjectures 1, 2, and3 of Atiyah and Jones in the affirmative.Theorem A.

For all k > 0, the natural inclusion ϑk : Mk →Ω3kSU(2) is ahomotopy equivalence through dimension q = q(k) = [k/2] −2.By a homotopy equivalence through dimension q(k) we mean that ϑk inducesan isomorphism πt(Mk)∼=πt(Ω3kSU(2)) = πt+3(SU(2)) for t ≤q(k).Thus,for example, the low-dimensional homotopy groups of Mk are finite. The state-ment in homotopy implies the weaker statement that ϑk induces an isomorphismHt(Mk; A) ∼= Ht(Ω3kSU(2); A) for t ≤q(k) and any coefficient group A. Here,the homology groups on the right are completely known.Actually, Theorem A is a corollary of the main topological result in [BHMM1].Theorem B.

For all k > 0, the inclusion ιk : Mk →Mk+1 is a homotopyequivalence through dimension q = q(k) = [k/2] −2.These topological results are actually formal consequences of our detailed geo-metric analysis of Mk. This is explained in the following section.

Lastly, it is likelythat the explicit bound q(k) can be improved as explained at the end of the nextsection.2. Outline of proofsA key tool in our approach is a theorem of Donaldson [D], that allows us to workin a purely holomorphic context and study a certain moduli space of holomorphicbundles.

To prove Theorems A and B we study this equivalent holomorphic moduli

THE ATIYAH-JONES CONJECTURE3space by following the program used in [MM1] and [MM2] to study the topologyof holomorphic maps from the Riemann sphere into flag manifolds. This shouldnot be surprising since these holomorphic mapping spaces correspond, by resultsof [Hu2] and [HM], to moduli spaces for SU(n)-monopoles.

The main idea is toconstruct a stratification of the entire moduli space by smooth manifolds, each withan orientable normal bundle, and each contained in the closure of higher dimen-sional strata. These strata are then organized into a filtration, and the resultingspectral sequence is analyzed to prove homology versions of Theorems A and B.More precisely, we proveTheorem C. For each k and all coefficient rings A, there are homology Lerayspectral sequences Er(Mk; A) converging to filtrations of H∗(Mk; A) with identifi-able E1 terms (see [BHMM1] for the precise statement).

Furthermore, the inclu-sion ιk : Mk →Mk+1 induces a map of spectral sequences ιk(r): Er(Mk; A) →Er(Mk+1; A). In particular, differentials in these spectral sequences are naturalwith respect to ιk(r).Next, we show that the map ιk induces an isomorphism of the E2 terms inTheorem C through dimension q(k) + 2.

Since differentials are natural, it followsthat ιk induces a homology equivalence through dimension q(k) + 1. This gives thehomology version of Theorem B, namely,Theorem D. For all k > 0 and all coefficients A, the inclusion ιk : Mk →Mk+1induces an isomorphism in homology (ιk)t : Ht(Mk; A) ∼= Ht(Mk+1; A) for t ≤q = q(k) + 1.Combined with the homology Taubes-Gravesen stability theorem, Theorem Dimplies the homology version of Theorem A.

However, as Mk and Ω3S3 are notsimply connected, Theorems A and B do not trivially follow from their homol-ogy analogs. Thus, we analyze the induced map on the universal covers (˜ϑk)t :H∗( fMk; Z) →Ht(eΩ3S3; Z) and show this is an isomorphism for t ≤q(k).

Thisnow implies Theorems A and B.Thus, to complete the outline of the proof we need to explain how we proveTheorem C. It is nontrivial to obtain a stratification for Mk that is suited to thistype of topological analysis. In fact, much of the work in [BHMM1] is devoted tounderstanding the geometry of Mk in sufficient detail so that we can constructsuch a stratification.

As mentioned above the starting point for this analysis is aresult of Donaldson [D] saying that the framed SU(2) instanton moduli space isequivalent to the moduli space of rank two holomorphic vector bundles on P2 withc1 = 0 that are trivial when restricted to a fixed line ℓwith a fixed trivializationthere.Moreover, as was noticed in [A] and [Hu1], it is convenient to performa birational transformation on the line ℓto obtain a surface ruled by lines, forexample, P1 × P1. Let Mk(M, S) denote the moduli space of isomorphism classesof rank 2 holomorphic bundles on M with c1 = 0 and c2 = k that have a fixedtrivialization on S. Then there are diffeomorphismsMk ≃Mk(P2, ℓ) ≃Mk(P1 × P1, ℓ1 ∨ℓ2).Thus, the geometry of Mk is described by isomorphism classes of certain framedsemistable holomorphic rank 2 vector bundles E over P1×P1.

It is known that suchE are trivial on almost all lines P1 × {x} and that the structure of E is determined

4C. P. BOYER, J. C. HURTUBISE, B. M. MANN, AND R. J. MILGRAMby its behaviour on neighbourhoods of a finite number of jumping lines P1 × {xi},lines over which the holomorphic structure of E is that of a sum of line bundlesO(h) ⊕O(−h) with h > 0.

In particular, if {∞} in the second P1 corresponds tothe line ℓ1, then the possible jumping lines are parameterized by the complex planeC ≃P1 −{∞}. At each jumping line there is an associated integer m called themultiplicity.

Let Qm denote the framed isomorphism classes (on a neighbourhoodof a jumping line) of framed jumps of multiplicity m. To each instanton ω is thusassociated an isomorphism class [Eω] of bundles Eω and, in turn, to each [Eω] isassociated a point in a labelled configuration space. Elements of this space aregiven by configurations of points zi in the complex plane C determined by a finitenumber of nontrivial jumping lines each labelled by elements li ∈Qmi.

Here thetotal charge k is the sum of the multiplicities.Thus, as shown in [Hu1], thereis a natural holomorphic projection Π : Mk →SP k(C) that associates to anyequivalence class of framed bundles [E] its divisor of jumping lines Pri=1 mizi inSP k(C). The fibre at a point z ∈SP k(C) with multiplicity (m1, .

. .

, mr) is theproduct Qm1 × · · · × Qmr where Qm denotes the space of equivalence classes offramed jumps of multiplicity m. This is the “pole and principal part” picture ofSegal-Gravesen [Gra].However, as each Qm is not necessarily a smooth manifold, this decomposition ofMk is yet not sufficiently fine to apply the techniques of [MM1, MM2]. Therefore, arefined local analysis of Qm is required.

To each multiplicity mi we associate finitesequences G = (h0, h1, . .

. , hl−1) of decreasing integers called “heights” such thatPl−1j=0 hj = mi.

We refer to these sequences G as “graphs” and show that to eachframed jumping line of multiplicity m we can associate a unique graph G. This isdone by setting, for each k,hk = dimsections of E ⊗O(−1) on the jumping line which extendto the kth formal neighbourhood of the line in P1 × P1.We thus obtain a decomposition Qm = FG FJG where each FJG is the collectionof framed jumps of a fixed graph type and the union is taken over all graphs G ofmultiplicity m. Most importantly, we proveTheorem E. Let G be a graph with heights h = h0 ≥h1 ≥· · · ≥hl−1 > hl = 0,length l, and multiplicity m = P hi. The space of framed jumps FJG with graph Gis a smooth complex manifold of complex dimension 2m + l.Let SG1,...,Gr denote the set of equivalence classes of framed holomorphic bundleswith r jumping lines (L1, .

. .

, Lr) with graphs G(r) = (G1, . .

. , Gr), respectively.Thus, we have the disjoint union decomposition Mk = F SG.

Furthermore, themap Π now restricts to a map ΠG(r) : SG(r) →DPr(C) with fibre FJG1 ×· · ·×FJGr.Here DPr(C) is the deleted product; that is, the space of r distinct unordered pointsin C, which is a smooth complex manifold of complex dimension r. From TheoremE followsTheorem F. SG(r) is a smooth complex variety of complex dimension 2k + l + r.We then prove that this decomposition of Mk by the SG submanifolds is a strat-ification of the type required. In particular, it is necessary to know the codimensionof each stratum (which is given in the last theorem), that there is a natural lex-icographical order on the graph types, and, with respect to that order, what the

THE ATIYAH-JONES CONJECTURE5intersection of the normal bundle of a fixed stratum is with the other strata. Inaddition to checking these facts we verify that our constructions are all compati-ble with the stabilization maps ιk in diagram (1.1).

After combining these resultsTheorem C follows as in [MM1, MM2].Finally, we note that the stability argument given here does not require explicitcalculation of differentials in the spectral sequence given in Theorem C but uses onlyinformation at the E1 level and naturality. In a sequel [BHMM2] we will examinethe structure of the E1 term and of the differentials much more carefully.

This willpermit us to both sharpen the explicit value of q(k) and, more importantly, obtainmuch more detailed information about H∗(Mk) above the range of stability.References[A]M. F. Atiyah, Instantons in two and four dimensions, Comm. Math.

Phys. 93 (1984),437–451.[AJ]M.

F. Atiyah and J. D. Jones, Topological aspects of Yang-Mills theory, Comm. Math.Phys.

61 (1978), 97–118. [BHMM1] C. P. Boyer, J. C. Hurtubise, B. M. Mann, and R. J. Milgram, The topology of instantonmoduli spaces.

I: The Atiyah-Jones conjecture, Ann. of Math.

(2), (to appear). [BHMM2], The topology of instanton moduli spaces.

II: The Toeplitz varieties, in prepa-ration.[BoMa]C. P. Boyer and B. M. Mann, Homology operations on instantons, J. Differential Geom.28 (1988), 423–465.[D]S.

K. Donaldson, Instantons and geometric invariant theory, Comm. Math.

Phys. 93(1984), 453–461.[Gra]J.

Gravesen, On the topology of spaces of holomorphic maps, Acta Math. 162 (1989),247–286.[Hu1]J.

C. Hurtubise, Instantons and jumping lines, Comm. Math.

Phys. 105 (1986), 107–122.

[Hu2], The classification of monopoles for the classical groups, Comm. Math.

Phys.120 (1989), 613–641.[HM]J. C. Hurtubise and M. K. Murray, On the construction of monopoles for the classicalgroups, Comm.

Math. Phys.

122 (1989), 35–89.[MM1]B. M. Mann and R. J. Milgram, Some spaces of holomorphic maps to complex Grass-mann manifolds, J. Differential Geom.

33 (1991), 301–324. [MM2], On the moduli space of SU(n) monopoles and holomorphic maps to flag man-ifolds, preprint, UNM and Stanford University, 1991.[T1]C.

H. Taubes, Path-connected Yang-Mills moduli spaces, J. Differential Geom. 19(1984), 337–392.

[T2], The stable topology of self-dual moduli spaces, J. Differential Geom. 29 (1989),163–230.Department of Mathematics and Statistics, University of New Mexico, Albu-querque, New Mexico 87131E-mail address: cboyer@gauss.unm.eduDepartment of Mathematics and Statistics, McGill University, Montr´eal, Qu´ebecH3G 1M8, CanadaE-mail address: hurtubis@gauss.math.mcgill.caDepartment of Mathematics and Statistics, University of New Mexico, Albu-querque, New Mexico 87131E-mail address: mann@gauss.unm.eduDepartment of Mathematics, Stanford University, Stanford, California 94305E-mail address: milgram@gauss.stanford.edu

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