On Casson-type instanton moduli spaces over negative definite four-manifolds
aa r X i v : . [ m a t h . G T ] N ov ON CASSON-TYPE INSTANTON MODULI SPACES OVERNEGATIVE DEFINITE FOUR-MANIFOLDS
ANDREW LOBBRAPHAEL ZENTNER
Abstract.
Recently Andrei Teleman considered instanton moduli spaces overnegative definite four-manifolds X with b ( X ) ≥
1. If b ( X ) is divisible byfour and b ( X ) = 1 a gauge-theoretic invariant can be defined; it is a countof flat connections modulo the gauge group. Our first result shows that ifsuch a moduli space is non-empty and the manifold admits a connected sumdecomposition X ∼ = X X then both b ( X ) and b ( X ) are divisible by four;this rules out a previously naturally appearing source of 4-manifolds with non-empty moduli space. We give in some detail a construction of negative definite4-manifolds which we expect will eventually provide examples of manifoldswith non-empty moduli space. Introduction
Recently Andrei Teleman considered moduli spaces of projectively anti-selfdualinstantons in certain Hermitian rank-2 bundles over a closed oriented 4-manifoldwith negative definite intersection form [12]. These play a role in his classificationprogram on Class VII surfaces [13][14]. However, in certain situations the instantonmoduli spaces involved consist of projectively flat connections and therefore havevery interesting topological implications. In this article we will study these ‘Casson-type’ moduli spaces.Suppose E → X is a Hermitian rank-2 bundle with first Chern-class a (minimal)characteristic vector w of the intersection form. In other words, it is the sum ofelements { e i } in H ( X ; Z ) which induce a basis of H ( X ; Z ) / Tors diagonalisingthe intersection form (because of Donaldson’s theorem [2]). Then for one possiblevalue of a strictly negative second Chern class c ( E ) the moduli space is compact(independently of the Riemannian metric). In particular, if the manifold has secondBetti-number b ( X ) divisible by 4 and first Betti-number b ( X ) = 1 the instantonmoduli space consists of projectively flat connections and has expected dimensionzero. This should be thought of as a ‘Casson-type’ moduli space because the ho-lonomy yields a surjection onto the space of SO (3) representations of π ( X ) withfixed Stiefel-Whitney class w = w ( mod e i can be Poincar´e dual to an element representable by a sphere, i.e. to an elementin the image of the Hurewicz homomorphism. Prasad and Yeung [10] constructedaspherical manifolds W which are rational-cohomology complex projective planes,generalisations of Mumford’s fake projective plane [9]. If W denotes this manifoldwith the opposite orientation, a natural candidate of a manifold for which themoduli space might be non-empty is given by the connected sum 4 W of 4 copies of W , and a candidate of a manifold for which the Casson-invariant can be defined is given by a ‘ring of 4 copies of W ’ (the last summand in the 4-fold connected sum4 W is taken a connected sum with the first).After recalling the gauge-theoretical situation considered in [12] we show that ifthe Casson-type moduli space is non-empty, then we cannot have a connected sumdecomposition X ∼ = X X unless both b ( X ) and b ( X ) are divisible by four.In particular the moduli space for the above mentioned 4 W - ring is empty.This result still leaves open the question of whether there is any X with anon-empty Casson-type moduli space. We give therefore in some detail a possi-ble construction of suitable 4-manifolds X (along with the correct representationsof π ( X )). We would like to point out that even though recent investigation leadsus to believe that the Casson-type invariant is vanishing [15], the Casson-type mod-uli space may still be non-empty and is interesting from a topological perspective.Our construction also suggests the possibility of considering Casson-type modulispaces for manifolds with boundary. Remark.
A similar moduli space and invariant has been defined by Ruberman andSaveliev for Z [ Z ] -homology Hopf surfaces, going back to work of Furuta and Ohta [5] , and for Z [ Z ] -homology 4-tori [11] . Our situation is simpler than their firstmentioned situation because of the absence of reducibles in the moduli space due tothe condition on b ( X ) . Acknowledgements
The first author thanks Simon Donaldson for useful conversations. The sec-ond author is grateful to Andrei Teleman for turning his interest to low-energyinstantons and for a stimulating conversation on them, and also wishes to expresshis gratitude to Stefan Bauer for helpful conversations. Both authors thank KimFrøyshov profusely for invaluable advice and ideas. We are also grateful to thereferee for the care taken in helping us substantially improve the article.1.
Donaldson theory on negative definite four-manifolds,low-energy instantons
After briefly recalling some general instanton gauge theory [4], and introducingour notations, we shall quickly turn to the special situation of ‘low-energy instan-tons’ over negative definite 4-manifolds mentioned in the introduction. We showthat the gauge-theoretical situation is indeed relatively simple, indicate a definitionof an invariant, and set up the correspondance of the moduli space to representationspaces of the fundamental group in SO (3).1.1. Connections.
Let X be a smooth Riemannian four-manifold and E → X a Hermitian rank-2 bundle on X . Let further a be a fixed unitary connection inthe associated determinant line bundle det ( E ) → X . We define A a ( E ) to be theaffine space of unitary connections on E which induce the fixed connection a in det ( E ). This is an affine space over Ω ( X ; su ( E )), the vector space of su ( E )-valuedone-forms on X . Let us denote by P the principal U (2) bundle of frames in E , andlet P be the bundle that is associated to P via the projection π : U (2) → P U (2), P = P × π P U (2). The space A ( P ) of connections in the P U (2) principal bun-dle P and the space A a ( E ) are naturally isomorphic. If we interpret a connection A ∈ A a ( E ) as a P U (2) connection via this isomorphism it is commonly called a pro-jective connection. The adjoint representation ad : SU (2) → SO ( su (2)) descends ASSON-TYPE MODULI SPACES OVER DEFINITE 4-MANIFOLDS 3 to a group isomorphim
P U (2) → SO ( su (2)). The associated real rank-3 bundle P × ad su (2) is just the bundle su ( E ) of traceless skew-symmetric endomorphismsof E . Thus the space A a ( E ) is also isomorphic to the space A ( su ( E )) of linearconnections in su ( E ) compatible with the metric. We shall write A ∈ A ( P ) forconnections in the P U (2) principal bundle and denote the associated connection in su ( E ) by the same symbol. Should we mean the unitary connection which inducesthe connection a in det( E ) we will write A a instead.Let G denote the group of automorphisms of E of determinant 1. It is called the‘gauge group’. This group equals the group of sections Γ( X ; P × Ad SU (2)), whereAd : U (2) → Aut( SU (2)) is given by conjugation. We shall write B ( E ) for thequotient space A ( P E ) / G . A connection is called reducible if its stabiliser underthe gauge group action equals the subgroup given by the centre Z / Z ( SU (2))which always operates trivially, otherwise irreducible . Equivalently, a connection A a is reducible if and only if there is a A a - parallel splitting of E into two propersubbundles.Let us point out that the characteristic classes of the bundle su ( E ) are given by w ( su ( E )) = c ( E ) ( mod p ( su ( E )) = − c ( E ) + c ( E ) . (1)1.2. Moduli space of anti-selfdual connections.
For a connection A ∈ A ( P )we consider the anti-selfduality equation F + A = 0 , (2)where F A denotes the curvature form of the connection A , and F + A its self-dual partwith respect to the Hodge-star operator defined by the Riemannian metric on X .The moduli space M ( E ) ⊆ B ( E ) of antiself-dual connections, M ( E ) = { A ∈ A ( P E ) (cid:12)(cid:12) F + A = 0 } / G is the central object of study in instanton gauge theory. This space is in generalnon-compact and there is a canonical “Uhlenbeck-compactification” of it. The anti-selfduality equations are elliptic, so Fredholm theory provides finite dimensional lo-cal models for the moduli space. The often problematic aspect of Donaldson theoryis the need to deal with reducible connections and with a non-trivial compactifi-cation. We will consider special situations where these problems do not occur.1.3. Low-energy instantons over negative definite four-manifolds.
We re-strict now our attention to smooth Riemannian four-manifolds X with b +2 ( X ) = 0and b ( X ) ≥
1. According to Donaldson’s theorem [2] the intersection form of sucha four-manifold is diagonal. Let { e i } be a set of elements in H ( X ; Z ) which inducea basis of H ( X ; Z ) / Tors diagonalising the intersection form.
Lemma 1.1. [12, section 4.2.1]
Suppose the Hermitian rank-2 bundle E → X has first Chern class c ( E ) = P e i and its second Chern class is strictly negative, c ( E ) < . Then E → X does not admit any topological decomposition E = L ⊕ K into the sum of two complex line bundles.Proof: Suppose E = L ⊕ K . Then c ( L ) = P l i e i and c ( K ) = P e i − P l i e i for some l i ∈ Z . Therefore, c ( E ) = c ( L )( c ( E ) − c ( L )) = X ( l i − l i ) ≥ . ANDREW LOBB RAPHAEL ZENTNER (cid:3)
Corollary 1.2.
Let E → X be as in the previous lemma. Then the moduli space M ( E ) does not admit reducibles. For a connection A ∈ A ( su ( E )) Chern-Weil theory gives the following formula:18 π ( k F − A k L ( X ) − k F + A k L ( X ) ) = − p ( su ( E )) = c ( E ) − c ( E ) (3)In particular, for anti-selfdual connections the left hand side of this equation isalways non-negative, and we can draw the following observation from the formula: Observation. [12, p. 1717]
1. For c ( E ) − / c ( E ) ∈ { , / , / , / } the mod-uli space M ( E ) is always compact, independently of the chosen metric or any gener-icity argument. In fact, the lower strata in the Uhlenbeck-compactification consist ofanti-selfdual connections in bundles E k with c ( E k ) = c ( E ) and c ( E k ) = c ( E ) − k for k ≥ .2. For c ( E ) = P e i we have c ( E ) = − b ( X ) . Thus, if b ( X ) ≡ mod and c ( E ) = − b ( X ) the moduli space M ( E ) will consist of projectively flat connec-tions only. We recall the expected dimension of the moduli space M ( E ). It is given by theformula d ( E ) = − p ( su ( E )) + 3( b ( X ) − b +2 ( X ) − d ( E ) ≥ b +2 ( X ) = 0, c ( E ) = P e i , and c ( E ) <
0, the latter condition assuring that we arein the favorable situation of Lemma 1.1.Interesting is the following special case of ‘Casson-type’ moduli spaces that weconsider from now on:
Proposition 1.3.
Let X be a negative definite Riemannian four-manifold withstrictly positive second Betti-number b ( X ) divisible by four, and b ( X ) = 1 . Let E → X be a Hermitian rank-2 bundle with c ( E ) = P e i and with c ( E ) = − / b ( X ) . Then the moduli space M ( E ) of projectively anti-selfdual connec-tions in E is compact and consists of irreducible projectively flat connections only,and is of expected dimension zero. After suitable perturbations a gauge theoretic invariant can be defined in thissituation: It is an algebraic count of a perturbed moduli space which consists of afinite number of points, the sign of each point is obtained by a natural orientationdetermined by the determinant line bundle of a family of elliptic operators. This hasbeen done in the meantime in [15], where it is shown that this invariant is actuallyzero. We would like to emphasise that the vanishing of this invariant doesn’t implyemptiness of the unperturbed moduli space that we shall investigate further here.1.4.
Flat connections, holonomy and representations of the fundamentalgroup.
Suppose we are in the situation that our moduli space M ( E ) consists offlat connections in su ( E ) → X , as for instance in the last proposition. Then wemust have p ( su ( E )) = 0 by Chern-Weil theory.The holonomy establishes a correspondance between flat connections in the ori-ented real rank-3 bundle V → X and representations of the fundamental group π ( X ) in SO (3) with a prescribed Stiefel-Whitney class. More precisely, let ρ : ASSON-TYPE MODULI SPACES OVER DEFINITE 4-MANIFOLDS 5 π ( X ) → SO (3) be a representation of the fundamental group. Let e X be theuniversal covering of X ; it is a π ( X ) principal bundle over X . We can form theassociated oriented rank-3-bundle V ρ := e X × ρ R . It admits a flat connection as it is a bundle associated to a principal bundle withdiscrete structure group. Therefore it has vanishing first Pontryagin class, p ( V ρ ) =0, by Chern-Weil theory. Its only other characteristic class [1] is its second Stiefel-Whitney class w ( V ρ ). Therefore we will say that the representation ρ has Stiefel-Whitney class w ∈ H ( X ; Z /
2) if w = w ( V ρ ). On the other hand, let V → X bean oriented real rank-3 bundle with a flat connection A . Then the holonomy of A along a path only depends up to homotopy on the path, and therefore induces arepresentation Hol ( A ) : π ( X ) → SO (3) = SO ( V | x ). In particular, the holonomydefines a reduction of the structure group to π ( X ), and the bundle can thereforebe reconstructed as V ∼ = V Hol ( A ) . In particular the representation Hol ( A ) hasStiefel-Whitney class w ( V Hol ( A ) ) = w ( V ).The moduli space M ( E ) has been obtained by quotienting the space of antiself-dual connections in A ( P E ) ∼ = A ( su ( E )) by the gauge group G . From the perspec-tive of the P U (2) connections in su ( E ) this gauge group is not the most naturalone. Instead, the group G := Γ( X ; P × Ad P U (2))is the natural group of automorphisms of connections in su ( E ). Not every element g ∈ G admits a lift to G ; instead, there is a natural exact sequence1 → G → G → H ( X ; Z / → . Quotienting by G has the advantage of a simpler discussion of reducibles, as dis-cussed above. Let us denote by M ( su ( E )) := { A ∈ A ( su ( E )) | F + A = 0 } / G the moduli space of anti-self dual connections in su ( E ) modulo the full gauge group G . Then there is a branched covering M ( E ) → M ( su ( E )) with ‘covering group’ H ( X ; Z / R w ( π ( X ); SO (3)) the space of representations of π ( X ) in SO (3) up to conjugation and of Stiefel-Whitney class w ∈ H ( X ; Z / Hol : M ( su ( E )) ∼ = → R w ( π ( X ); SO (3)) , where w = w ( su ( E )). In particular, M ( E ) surjects onto R w ( π ( X ); SO (3)).2. Representations of the fundamental group in SO (3) and thevanishing result We will use the above derived relation of the ‘Casson-type moduli space’ M ( E )to the representation space R w ( π ( X ); SO (3)) to obtain the vanishing result whichis mentioned in the introduction. ANDREW LOBB RAPHAEL ZENTNER
Flat SO (3) bundles. The above construction of the bundle V ρ associated toa representation ρ : π ( X ) → SO (3) is functorial in the following sense: Lemma 2.1.
Suppose we have a map f : W → X between topological spaces, and ρ : π ( X ) → SO (3) a representation of the fundamental group of X . Then there isa natural isomorphism f ∗ V ρ ∼ = V ρ ◦ f ∗ (4) between the pull-back of the bundle V ρ via f and the bundle V ρ ◦ f ∗ → W , where f ∗ : π ( W ) → π ( X ) is the map induced by f on the fundamental groups.Proof: We have a commutative diagram f W e XW X, ✲ e f ❄ ❄✲ f where the vertical maps are the universal coverings, and where e f is the unique mapturning the diagram commutative (we work in the category of pointed topologicalspaces here). It is elementary to check that the map e f is equivariant with respectto the action of π ( W ), where this group acts on e X via f ∗ : π ( W ) → π ( X ) andthe deck transformation group of e X . The claimed isomorphism follows then fromnaturality of the associated bundle construction. (cid:3) Proposition 2.2.
Suppose the four-manifold X splits along the connected 3-mani-fold Y as X = X ∪ Y X into two four-manifolds X and X . Then any represen-tation ρ : π ( X ) → SO (3) induces representations ρ i : π ( X i ) → SO (3) via ρ ◦ ( j i ) ∗ where the map j i : X i ֒ → X is the inclusion. For these representations we have V ρ | X i = V ρ i . (5) Conversely, given representations ρ i : π ( X i ) → SO (3) such that ρ ◦ ( k ) ∗ = ρ ◦ ( k ) ∗ : π ( Y ) → SO (3) , where k i : Y ֒ → X i denotes the inclusion, there is arepresentation ρ : π ( X ) → SO (3) inducing ρ and ρ via the respective restrictions.Proof: This follows from the Theorem of Seifert and van Kampen and the lemmaabove or, equivalently, by gluing connections. (cid:3)
Vanishing results for Casson-type moduli spaces.Proposition 2.3.
Let X be a four-manifold with b +2 ( X ) = 0 , and let w ∈ H ( X ; Z / be P e i ( mod . Suppose there is a representation ρ : π ( X ) → SO (3) with fixedsecond Stiefel-Whitney class w . Then none of the Poincar´e dual of the basis ele-ments e i is in the image of the Hurewicz-homomorphism h : π ( X ) → H ( X ; Z ) .Proof [12, p. 1718] : Suppose we have a map f : S → X such that P D ( e i ) = f ∗ [ S ], where [ S ] ∈ H ( S ; Z ) denotes the fundamental cycle of S , and P D ( e i )denotes the Poincar´e dual of e i . Then we have h w, f ∗ [ S ] i ≡ h X e j , P D ( e i ) i = e i = − mod . (6) ASSON-TYPE MODULI SPACES OVER DEFINITE 4-MANIFOLDS 7
On the other hand, by naturality of the cohomology-homology pairing, we get h w, f ∗ [ S ] i = h w ( V ρ ) , f ∗ [ S ] i = h f ∗ w ( V ρ ) , [ S ] i . (7)But the above Lemma 2.1 implies that f ∗ w ( V ρ ) = w ( f ∗ V ρ ) = w ( V ρ ◦ f ∗ ). As S has trivial fundamental group the bundle V ρ ◦ f ∗ is clearly the trivial bundle, so theleft hand side of equation (7) must be zero modulo 2, a contradiction to equation(6). (cid:3) Remark.
By Hopf ’s theorem on the cokernel of the Hurewicz-homomorphism, ex-pressed in the exact sequence π ( X ) → H ( X ; Z ) → H ( π ( X ); Z ) → , the fundamental group has to have non-trivial second homology in order to obtaina non-empty Casson-type moduli space. This proposition gives a topological significance of the zero-energy instantons: Ifthe moduli space is non-empty then the elements
P D ( e i ) are not representable byspheres! One might wonder whether there exists any four-manifold where the ele-ments P D ( e i ) are not representable by spheres. Certainly this cannot be a simplyconnected four-manifold because of the Hurewicz-isomorphism theorem. Interest-ingly, the answer is affirmative. Generalising Mumford’s fake projective plane [9],Prasad and Yeung have constructed manifolds with the rational cohomology of thecomplex projective space CP whose universal cover is the unit ball in C [10]. Sucha manifold W is therefore an Eilenberg-MacLane space K ( π ( W ) , Z be the four-manifold that we obtain from the connected sum of four W , where we do again a connected sum of the last summand with the first. The soobtained “4- W -ring” is diffeomorphic to Z := W W W W S × S =: 4 W S × S . This manifold has negative definite intersection form and has Betti-numbers b ( Z ) =1 and b ( Z ) = 4. In addition, no element of H ( Z, Z ) is representable by a 2-sphere,so we get no obstruction to non-emptiness from Proposition 2.3. Thus the four-manifold Z is a prototype of a four-manifold on which to consider the moduli spaceof P U (2) instantons associated to the bundle E → Z with c ( E ) = P e i and c ( E ) = − b ( X ) (and therefore of representations of π ( X ) → SO (3) with fixedStiefel-Whitney class w = P e i ( mod Theorem 2.4.
Let X be a smooth closed negative definite four-manifold. If there isa representation ρ : π ( X ) → SO (3) with Stiefel-Whitney class w := P e i ( mod ,then the second Betti-number b ( X ) must be divisible by four.Proof: The bundle V ρ has w ( V ρ ) = w and vanishing first Pontryagin-class p ( V ρ ) = 0 because this bundle admits a flat connection. Now the Dold-Whitneytheorem [1] states that the second Stiefel-Whitney class w and the first Pontryaginclass p of any oriented real rank-3 bundle satisfy the equationP-Sq( w ) = p ( mod . Here P-Sq : H ( X ; Z / → H ( X ; Z /
4) denotes the Pontryagin square, a lift of thecup-product squaring H ( X ; Z / → H ( X ; Z /
2) to the coefficient group Z /
4. If
ANDREW LOBB RAPHAEL ZENTNER the class v ∈ H ( X ; Z /
2) is the mod-2 reduction of an integral class c ∈ H ( X ; Z )then the Pontryagin square is simply the mod-4 reduction of the square of c , i.e.P-Sq( v ) = c ( mod . In our case the Dold-Whitney theorem thus implies that0 = P-Sq( w ) = X e i = − b ( X ) ( mod . (cid:3) Hence we obtain the following
Theorem 2.5.
Let X be a four-manifold with negative definite intersection formand suppose it admits a connected sum decomposition X X . Suppose ρ : π ( X ) → SO (3) is a representation of the fundamental group of X with fixed Stiefel-Whitneyclass w = P e i ( mod . Then both b ( X ) and b ( X ) must be divisible by four.Proof: Note first that the intersection form of both X and X must be diagonal.This follows from Eichler’s theorem on unique decomposition of symmetric definiteforms over Z , see [6]. Therefore the basis vectors { e i } of H ( X ; Z ) are simply givenby the union of basis vectors { f i } of H ( X ; Z ), diagonalising the intersection formof X , and basis vectors { g i } of H ( X ; Z ), diagonalising the intersection form of X .Note that π ( X i \ B ) ∼ = π ( X i ). The above Proposition 2.2 now applies yieldingrepresentations ρ i : π ( X i ) → SO (3). Its second Stiefel-Whitney class computes,using the above equation (5), w ( V ρ ) = w ( V ρ ) | X \ B = X f i ( mod , and likewise for w ( V ρ ). The above theorem therefore concludes the proof. (cid:3) Corollary 2.6.
This implies that the above considered manifold Z = 4 W S × S does not admit a representation ρ : π ( X ) → SO (3) with Stiefel-Whitney class beingthe mod-2 reduction of the sum of basis elements diagonalising the intersection form. Remark.
As a ‘converse’ to the above vanishing theorem, suppose we are givena connected sum X = X X and representations ρ i : π ( X i ) → SO (3) with thedesired Stiefel-Whitney classes on X i , i = 1 , . According to Proposition 2.2, weobtain the representation ρ = ρ ∗ ρ : π ( X ) → SO (3) which has the desiredStiefel-Whitney class. This is in contrast to well-known vanishing theorems forconnected sums of manifolds with b +2 ( X i ) > , i = 1 , as in [4, Theorem 9.3.4 and,in particular, Proposition 9.3.7] . Constructing -manifolds with non-empty Casson-type modulispace There is much interest in the relationship between the fundamental group of a4-manifold and its intersection form. The Casson-type invariant considered in thispaper gives rise to the natural question of whether there exists any X with non-empty Casson-type moduli space. In this section we describe a construc-tion that we hope will provide the first examples of such manifolds, by indicatinghow to construct non-empty representation spaces R w ( π ( X ); SO (3)). ASSON-TYPE MODULI SPACES OVER DEFINITE 4-MANIFOLDS 9
Immersed -links and negative-definite -manifolds. Let ˜ L = ` m S → S be a smooth immersion of m L occur with negative sign and between two branches of the same component of˜ L . Suppose there are n self-intersections. Blowing up n times and taking the propertransform we obtain an m -component embedded link L = ` m S ֒ → n CP .Each component of L intersects each exceptional sphere of n CP either at nopoints or at one point positively and at one point negatively (this is because eachintersection point of ˜ L occurred within a single component and with negative sign).Hence each component of L is trivial homologically and so the embedding of L extends to a D -neighbourhood.We do surgery on L by removing L × D and gluing in ` m D × S . Call theresulting 4-manifold X . The construction of X was suggested by Kim Frøyshov. Itturns out to be very suited to our purposes; we have Lemma 3.1. (1) H ( X ; Z ) = ⊕ m Z . (2) H ( X ; Z ) = ⊕ n Z . (3) There is a basis for H ( X ; Z ) with each element represented by an embeddedtorus T ֒ → X . (4) The intersection form of X is diagonal and negative definite.Proof: Let Y = n CP \ ( L × D ) be the complement of the link L . Then n CP = Y ∪ m − handles ∪ m 4 − handles ,X = Y ∪ m − handles ∪ m 4 − handles . Hence • χ ( n CP ) − χ ( X ) = 2 m • H ( X ; Z ) = H ( Y ; Z ) • H ( Y ; Z ) ⊆ ⊕ m Z since n CP is simply connected.So we shall be done if we can find n embedded tori in X which are pairwise disjointand which each have self-intersection −
1. Figure 1 shows how to find these tori.Working inside n CP , each exceptional sphere E intersects L transversely in twopoints. Connect these two points by a path on L . The D -neighbourhood of L pulls back to a trivial D -bundle over the path. The fibres over the two endpointscan be identified with neighbourhoods of these two points in E . Removing theseneighbourhoods from E we get a sphere with two discs removed and we take theunion of this with the S boundaries of all the fibres of the D -bundle over thepath.This gives a torus which has self-intersection −
1, and we can certainly choosepaths on L for each exceptional sphere which are disjoint. (cid:3) We have shown how to associate to a given immersed 2-link ˜ L = ` m S → S with only negative self-intersections and disjoint components, a smooth 4-manifold X ˜ L which is diagonal and negative definite, with basis elements of H ( X ˜ L ; Z ) rep-resented by embedded tori. PSfrag replacements Exceptional sphere E Path on L Boundary of neighbourhood of path on L Figure 1.
A torus representing a basis element of H ( X ; Z ).3.2. SO (3) representations of π and presentations of -links. Using thesame notation as in the previous subsection, we give a method to describe links ˜ L that come with representations π ( X ˜ L ) → SO (3) with the correct Stiefel-Whitneyclass w = P e i ( mod 2). This method may not at first appear entirely general,but we show that if there is such a link ˜ L then it must admit a description of thisform.We start by giving a lemma, which follows from basic relative Morse theory: Lemma 3.2.
Any closed immersed surface in S admits a movie description inwhich the movie moves occur in the following order: (1) 0 -handles (circle creation). (2) Simple crossing changes (see Figure 3). (3)
Ribbon-type Reidemeister moves of type II (see Figure 4). (4)
Ribbon-type -handle addition (see Figure 4).After the ribbon-type -handle additions there remains a diagram of an unlinkand the only handle attachments left to do are -handle attachments (circle anni-hilation). (cid:3) Representations of the fundamental group and ribbon presentations of -links. We now explain how to describe a representation π ( n CP \ L ) → SO (3) from adecorated presentation of the immersed link ˜ L .For notation, let ˜ h : S → R be a height function corresponding to Lemma 3.2with exactly 2 critical points that restricts to a Morse function on ˜ L such that all the i -handles of ˜ L occur in ˜ h − ( − i ) and the self-intersections of ˜ L occur in ˜ h − ( − / h : n CP → R with one maximumand one minimum, and n index 2 critical points. These index 2 critical points all ASSON-TYPE MODULI SPACES OVER DEFINITE 4-MANIFOLDS 11
PSfrag replacements x x k y y l . . .. . .. . .. . . Figure 2.
The 0-handles of a 2-knot with 4 k negative self-intersections. PSfrag replacements − handles sprout ribbonsRibbons are allowedto overcross or undercrosseach other and0 − handlesRibbon − type 1 − handle addition y l . . . Figure 3. By simple crossing change we mean doing a Reidemeis-ter 2 move between two 0-crossing diagrams of the unknot and thenperforming a crossing change at one of the crossings we have in-troduced.occur at h − ( − / L with the i -handles of L occurring in h − ( − i ). We can use the same movie of ˜ L to describe the embedding of L .Recall that π ( X ˜ L ) = π ( n CP \ L ). We compute π ( n CP \ L ) using the VanKampen theorem. First note that h − ([ − / , ∞ )) \ L is the boundary connect sumof n copies of the complement of 2 fibres in the D -bundle over S of Euler class −
1, and l copies of D \ D where the D with ∂D ⊂ ∂D is trivially embedded.Here n is the number of self-intersections of ˜ L (and hence the number of blow-upsrequired on the way to constructing X ˜ L ) and l is the number of extra 0-handles PSfrag replacements 0 − handles sprout ribbons Ribbons are allowedto overcross or undercrosseach other and0 − handlesRibbon − type 1 − handle addition y l . . . Figure 4.
Ribbon-type moves in a movie presentation of an em-bedded surface in 4-space.used in the movie presentation of ˜ L satisfying Lemma 3.2. Since by assumption X ˜ L has a non-empty Casson-type moduli space and dim H ( X ˜ L ; Z ) = n , we can write n = 4 k by Theorem 2.4.The boundary of h − ([ − / , ∞ )) \ L is shown as the complement of the link inFigure 2, with a point at infinity which we fix as the basepoint.It is easy to compute that π ( h − ([ − / , ∞ )) \ L ) is the free (non-abelian) groupon 4 k + l generators. We fix representatives of a basis for this group as simple loopscoming down from infinity, linking the relevant circle by small meridians and head-ing back up again. For each of the 4 k generators coming from the blowups we allowourselves two representatives - one for each circle. Note that our representativeslive in the boundary of h − ([ − / , ∞ )) \ L .To get the space h − ([ − / , ∞ )) \ L we attach the complements of some 1-handlesto h − ([ − / , ∞ )) \ L . What this means is that for every 1-handle of L , we gluea D \ D to h − ([ − / , ∞ )) \ L , via a homeomorphism of ( D \ ( D ∪ D )) ⊆ ( S \ S ) = ∂ ( D \ D ) with a subset of ∂ ( h − ( − / \ L ). (All discs in thisdiscussion are trivially embedded).Since π ( D \ ( D ∪ D )) = Z × Z , π ( D \ D ) = Z , and the map on π inducedby inclusion is onto, the Van Kampen theorem tells us that adding the complementof a 1-handle adds a single, possibly trivial, relation to π . In other words, we ASSON-TYPE MODULI SPACES OVER DEFINITE 4-MANIFOLDS 13 obtain a presentation of π ( h − ([ − / , ∞ )) \ L ) with 4 k + l generators and as manyrelators as there are 1-handles.Since we obtain n CP \ L from h − ([ − / , ∞ )) \ L by gluing on the complementof some trivially embedded D ’s (one for each 2-handle of L ) in D , it followsthat π ( n CP \ L ) = π ( h − ([ − / , ∞ )) \ L ). Hence we have a presentationof π ( n CP \ L ). Now by assumption, X ˜ L has a non-empty Casson-type modulispace, so we choose some representation ρ : π ( n CP \ L ) = π ( X ˜ L ) → SO (3) thathas the correct associated characteristic classes. Each generator of the presentationis associated to some circle or Hopf link in Figure 2. We decorate each circle orHopf link with the image of the associated generator under ρ . We call these images x , x , . . . , x k , y , y , . . . , y l ∈ SO (3).Each 1-handle complement that we attach appears in the movie of ˜ L as a ribbon-type 1-handle addition as illustrated in Figure 4. Once we have added each ribbon-type handle then by assumption we have an unlink.3.2.2. Representations of the fundamental group and a singular link diagram.
Wenow reformulate the existence of ρ : π ( X ˜ L ) → SO (3) in terms of properties of themovie description of ˜ L and the decoration by x , . . . , x k , y , . . . , y l ∈ SO (3). Definition 3.3.
A singular link diagram G is given by • starting with the link diagram Figure 2 • adding the cores of each -handle of ˜ L .(For an example see Figure 6). Remark.
We could recover the full immersion ˜ L from G by adding a framing toeach -handle core in G , describing how to thicken the cores to the full -handles. Lemma 3.4.
These two statements are equivalent: (1)
Each component of ˜ L has genus . (2) Suppose two circles of Figure 2 are joined by three paths of -handle cores l , l , l in the singular diagram G . If l , l , l meet the first circle in threepoints that go clockwise (respectively anticlockwise) around the circle, then l , l , l must meet the second circle in three points that go anticlockwise(respectively clockwise) around that circle. Lemma 3.5.
These two statements are equivalent: (1)
Self-intersections of ˜ L only occur within a component and not between twocomponents of the preimage of ˜ L . (2) The singular diagram G describes an obvious singular link in R . Given aHopf link in Figure 2, we require that the two circles comprising it are partof the same component in this singular link. The proofs of Lemmas 3.4 and 3.5 are left as an exercise.
Lemma 3.6.
These two statements are equivalent: (1)
The representation π ( h − ([ − / , ∞ )) \ L ) → SO (3) determined by the labelling x , x , . . . , x k , y , y , . . . , y l ∈ SO (3) factorsthrough π ( n CP \ L ) . (2) Each circle in Figure 2 bounds an obvious oriented disc which has no dou-ble points when projected to the plane of the diagram. Consider a core ofa -handle A in the singular link diagram G . Suppose A connects circlesdecorated by SO (3) elements g and h , and that the arc, given the orienta-tion from g to h , intersects discs bounded by circles which are decorated byelements g , g , . . . , g m . Define the element C ( A ) = ( Q m g ± i ) , where the ± index is the sign of the intersection of the arc with the disc. We require h = C ( A ) gC ( A ) − . Proof.
The condition that the given representation π ( h − ([ − / , ∞ )) \ L ) → SO (3)factors through π ( n CP \ L )is equivalent to the representation killing the relators (coming from each 1-handleof ˜ L ) in the presentation of π ( n CP \ L ) discussed above.The calculation of the relators is illustrated in Figure 5. (cid:3) Lemma 3.7.
These two statements are equivalent: (1)
The representation ρ : π ( X ˜ L ) → SO (3) has the correct Stiefel-Whitneyclass w = P e i ( mod . (2) • For each Hopf link of Figure 2, choose a path of cores of -handlesin the singular diagram G which connects the circles of the Hopf link.(Such a path exists by Lemma 3.5).Say it consists of cores A , A , . . . , A m . We order and orient thesecores so that the start point of A and the end point of A m are ondifferent components of the Hopf link and the end point of A i is on thesame circle of Figure 2 as the start point of A i +1 for ≤ i ≤ m − .Write g for the element decorating the Hopf link. Then we require that m Y C ( A i ) = 1 , g. • The elements x , . . . , x k ∈ SO (3) are each conjugate to the element diag (1 , − , − (in other words each element x i is a rotation by π radians).Proof. The condition that the representation ρ : π ( X ˜ L ) → SO (3) has the correctStiefel-Whitney class says that: w ( i ∗ ρ ) = i ∗ ( w ( ρ )) = 0 ∈ H ( T ; Z /
2) = Z / , for the representative i : T → X of each basis element of H ( X ; Z ). ASSON-TYPE MODULI SPACES OVER DEFINITE 4-MANIFOLDS 15
PSfrag replacements ∞ g ˜ gg g Figure 5.
This diagram shows the situation just before the addi-tion of a 1-handle, which will take place within the dotted circle.We have indicated 4 generators of π ( n CP \ L ). By the VanKampen theorem, adding the 1-handle imposes the relation thatthe rightmost generator is a conjugate of the leftmost generator asin item 4 of our checklist. Thinking of the ribbon as a thickenedarc, we note that this calculation does not depend on whether thearc is locally knotted, but only on the order in and parity withwhich it intersects the discs bounded by the 0-handles. Also, sincea small loop encircling both strands of a ribbon clearly bounds adisc in h − ([ − / , ∞ )) \ L , it is also immaterial how the arcs linkeach other.By naturality, this means that the map ρ ◦ i : Z ⊕ Z = π ( T ) → SO (3) has togive the non-trivial flat bundle over T . Given a basis for π ( T ) this is equivalentto asking that ρ ◦ i sends each of the two basis elements to rotations by π , but aroundorthogonal axes. For each basis element of H ( X ˜ L ; Z ), there is an associated Hopflink in Figure 2. Say the Hopf link is decorated by g ∈ SO (3) and there is a pathconnecting the two components of the Hopf link as in Lemma 3.5.Consider the loop which we gave as a generator of π ( h − ([ − / , ∞ )) \ L ) cor-responding to the Hopf link, and a loop based at ∞ which goes down to the pathand follows it around until returning to the Hopf link and then returns back up to ∞ . This gives two basis elements for π of a T representing the basis element of H ( X ˜ L ; Z ). The former is sent to g by ρ ◦ i and the latter is sent to Q m C ( A i ). Sincenecessarily Q m C ( A i ) commutes with g , the requirement that Q m C ( A i ) = 1 , g , en-sures that Q m C ( A i ) is a rotation by π around an axis orthogonal to that of g . (cid:3) If we can find a presentation of some ˜ L with decoration by some x , . . . , x k , y , . . . , y l ∈ SO (3) satisfying the conditions of Lemmas 3.4, 3.5, 3.6, and 3.7, then we have seenthat we can construct a negative definite 4-manifold X ˜ L with non-empty Casson-type moduli space. In particular we have exhibited a particular representation π ( X ˜ L ) → SO (3)which has the required associated Stiefel-Whitney class.Giving such a presentation of ˜ L is equivalent to giving first the singular linkdiagram G and then giving a framing to the cores of each 1-handle. Therefore wehave the following: Theorem 3.8.
Suppose we give a singular link diagram G in the sense of Definition3.3, starting with Figure 2 and then adding arcs which begin and end at points ofFigure 2. Further suppose that there is a decoration of G by x , . . . , x k , y , . . . , y l ∈ SO (3) that satisfies the conditions on the singular link diagrams given as the latterstatements of Lemmas 3.4, 3.5, 3.6, and 3.7.Then, if there exists a framing of the arcs of G such that the corresponding -handle additions to Figure 2 gives a diagram of a trivial link, there exists a -manifold with non-empty Casson-type moduli space. (cid:3) A partial example.
The symmetry group on 4 elements S can be embeddedin SO (3) as the rotational symmetry group of a cube. Under this embedding, allelements of order = 2 are taken to rotations by π around some axis.In Figure 6 we have given an example of a diagram (of labelled Hopf links,simple circles, and arcs) satisfying all the conditions of Theorem 3.8. The groupelement decorations of the simple circles and the Hopf links are given in the cyclenotation for S ֒ → SO (3). If we can find a way to add more arcs, each satisfyingthe conditions of Theorem 3.8 (the Steifel-Whitney condition of Lemma 3.7 hasalready been satisfied in the diagram) such that when we replace each arc by aribbon we get the unlink, then we will have described an immersion ˜ L → S suchthat X ˜ L has non-empty Casson-type moduli space. Remark.
Note that each component of Figure 6 (after replacing each arc by ablackboard-framed -handle) is a smoothly slice knot. In fact in this case, more is true:
Proposition 3.9.
If Figure 6 is an intermediary diagram of a movie presenta-tion of an immersed ˜ L satisfying Theorem 3.8, then X ˜ L has exactly point in therepresentation space R w ( π ( X ); SO (3)) . Remark.
Recent discoveries [15] have indicated that the invariant defined as thesigned count of the Casson moduli space may always be . Results such as Propo-sition 3.9 are still valuable as they may be useful in showing that links are not slice(for more in this direction see [8] ).Proof. A representation ρ : π ( X ˜ L ) → SO (3) is determined by the decorationof the four Hopf links by elements of SO (3). We will see that there is only onepossible decoration up to conjugation.Suppose that we have some new decoration satisfying Theorem 3.8. Call thedecorating elements of SO (3) T L, T R, BL, BR where the initials stand for T op, B ottom, L eft, R ight. Each of the four decorations is a rotation by π around someaxis, so each element is equivalent to a choice of axis, and we use the same labelsfor these axes. By Lemma 3.7, we must have T L perpendicular to BL and T R perpendicular to BR . ASSON-TYPE MODULI SPACES OVER DEFINITE 4-MANIFOLDS 17
PSfrag replacements (12)(34) (14)(23)(24)
Figure 6.
An example of what a partial diagrammatic descriptionof a suitable immersion ˜ L → S may look like.There is an arc connecting T L to BL . By condition Lemma 3.6 we can interpretthis as meaning that the unique axis perpendicular to both T R and BR lies in thesame plane as T L and BL and is at an angle of π/ T R and BR , which implies that the axis perpendicularto T L and BL is in the same plane as T R and BR and at an angle of π/ SO (3). Hence, up to conjugation, thereis exactly 1 representation ρ : π ( X ˜ L ) → SO (3) of the correct characteristic class. (cid:3) References [1] A. Dold, H. Whitney,
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Mathematics Department, Imperial College London, London SW11 7AZ, UK
E-mail address : [email protected] Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, 33501 Bielefeld, Germany
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