Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants
HHYPERBOLIC FOUR-MANIFOLDS WITH VANISHINGSEIBERG-WITTEN INVARIANTS
IAN AGOL AND FRANCESCO LIN
Abstract.
We show the existence of hyperbolic 4-manifolds with van-ishing Seiberg-Witten invariants, addressing a conjecture of Claude Le-Brun. This is achieved by showing, using results in geometric and arith-metic group theory, that certain hyperbolic 4-manifolds contain L -spacesas hypersurfaces. Introduction
In [9, Conjecture 1 . § Theorem 1.1.
There exist closed arithmetic hyperbolic 4-manifolds withvanishing Seiberg-Witten invariants.
In the statement, we consider all possible Seiberg-Witten invariants com-ing from evaluating elements of the cohomology ring Λ ∗ H ( X ; Z ) ⊗ Z [ U ]of the space of configurations. Theorem 1.1 is proved by exhibiting hyper-bolic 4-manifolds admitting separating L -spaces, using the main result of[6]; under mild additional conditions, this implies that such manifolds ad-mit finite covers with vanishing Seiberg-Witten invariants. Our constructionwill show in fact that there are infinitely many commensurability classes ofarithmetic hyperbolic 4-manifolds containing representatives with vanishingSeiberg-Witten invariants. Furthermore, by interbreeding as in [4], one canalso obtain non-arithmetic examples. Acknowledgements.
The first author was funded by a Simons Investigatoraward. The second author was partially funded by NSF grant DMS-1807242.We thank Alan Reid for comments on an earlier draft. a r X i v : . [ m a t h . G T ] D ec IAN AGOL AND FRANCESCO LIN A vanishing criterion for the Seiberg-Witten invariants
We discuss a vanishing result for the Seiberg-Witten invariants of four-manifolds containing a separating hypersurface. This is well-known to ex-perts, but the exact form we will need is only implicitly stated in [7], so wewill point it out for the reader’s convenience. Most of our discussion is basedon formal properties of the invariants, and we will follow closely follow theexposition of [7, Chapter 3].Consider a spin c structure s X on a closed oriented 4-manifold X . Fora cohomology class u ∈ Λ ∗ H ( Y ; Z ) ⊗ Z [ U ], we define the Seiberg-Witteninvariant m ( u | X, s X ) to be the evaluation of u on the moduli space of so-lutions to the Seiberg-Witten equations. This is a topological invariantprovided that b +2 ≥
2. The latter is not a restrictive assumption in ourcase; hyperbolic 4-manifolds have signature zero by [2, Theorem 3] and theHirzebruch signature formula. Hence χ ( X ) = 2(1 − b ( X ) + b +2 ( X )) . If b +2 ( X ) ≤
1, we would have χ ( X ) ≤
4; on the other hand, in all knownexamples of closed orientable hyperbolic 4-manifolds χ ≥
16 [14, 11] (recallthat by Gauss-Bonnet, volume and Euler characteristic are proportional).We discuss a vanishing criterion for m ( u | X, s X ). Let Y be a closed, ori-ented three-manifold. To this, in [7, Section 3.1] it is defined for each spin c structure s on Y the monopole Floer homology groups fitting in the exacttriangle of graded Z [ U ]-modules(1) · · · −→ HM ∗ ( Y, s ) i ∗ −→ (cid:100) HM ∗ ( Y, s ) j ∗ −→ (cid:100) HM ∗ ( Y, s ) p ∗ −→ HM ∗ ( Y, s ) −→ · · · where U has degree − U with thecorresponding capping operation in homology). The reduced Floer group HM ∗ ( Y, s ) is defined to be the image of j ∗ in (cid:100) HM ∗ ( Y, s ) [7, Definition 3.6.3].We will be particularly interested in the case in which Y is a rational ho-mology sphere. In this case we have an identification of Z [ U ]-modules (upto grading shift) with Laurent series [7, Proposition 35.3.1] HM ∗ ( Y, s ) ∼ = Z [ U − , U ] . Definition 2.1 ([8]) . We say that a rational homology sphere Y is an L - space if, up to grading shift, (cid:100) HM ∗ ( Y, s ) = Z [ U ] as Z [ U ]-modules for all spin c structures s .As the map p ∗ in equation (1) is an isomorphism in degrees low enough[7, Section 22 . L -space HM ∗ ( Y, s ) = 0 for all spin c structures s Proposition 2.2.
Let X be a four-manifold given as X = X ∪ Y X . Sup-pose that the separating hypersurface Y is an L -space (so that in particular b ( Y ) = 0 ), and that b +2 ( X i ) ≥ . Then all the Seiberg-Witten invariants of X vanish. YPERBOLIC FOUR-MANIFOLDS WITH VANISHING SEIBERG-WITTEN INVARIANTS3
Remark 2.3.
A simpler to state vanishing criterion is the following: if b ( X ) = 0 and b +2 ( X ) is even, then all Seiberg-Witten invariants are zero.In fact, under this assumption all Seiberg-Witten moduli spaces are odddimensional [7, Theorem 1.4.4], while all classes in our cohomology ring areeven dimensional. On the other hand, we are not aware of examples ofhyperbolic 4-manifolds satisfying these conditions. Proof of Proposition 2.2.
All we need to do is to discuss the results of [7,Chapter 3] while keeping track of the specific spin c structures. First ofall, notice that as b ( Y ) = 0, a spin c structure s X on X is determinedby the restrictions s i = s X | X i . This follows from the injectivity of themap H ( X ; Z ) → H ( X ; Z ) ⊕ H ( X ; Z ) in the Mayer-Vietoris sequence,and the fact the these groups classify spin c structures. Let s = s X | Y . Itis sufficient to show that m ( u | X, s X ) = 0 for classes u = u u where u i is a cohomology class in the configuration space of X i . Recall from [7,Section 3.4] that a cobordism W from Y to Y induces a map in homologyfitting with the exact triangle; furthermore, if b +2 ( W ) ≥
1, we have that HM ∗ ( u | W, s ) = 0 [7, Proposition 3 . . ψ ( u | X , s ) ∈ (cid:100) HM ∗ ( Y, s ) obtained as follows: let W be the cobordism obtained from X by removing a ball, and consider theinduced map (cid:100) HM ∗ ( u | W , s ) : (cid:100) HM ∗ ( S ) = Z [ U ] → (cid:100) HM ∗ ( Y, s ) . Then ψ ( u | X , s ) = (cid:100) HM ∗ ( u | W , s )(1). On the other hand, we have thecommutative diagram (cid:100) HM ∗ ( S ) HM ∗ ( S ) (cid:100) HM ∗ ( Y, s ) HM ∗ ( Y, s ) p ∗ p ∗ (cid:100) HM ∗ ( u | W , s ) HM ∗ ( u | W , s )and as b +2 ( W ) ≥
1, the vertical map on the right vanishes; in turn, thisimplies that ψ ( u | X , s ) ∈ ker( p ∗ ) = HM ∗ ( Y, s ). Similarly, using the map in-duced in cohomology by W , we obtain an element ψ ( u | X , s ) ∈ HM ∗ ( − Y, s );this last group is by Poincar´e duality identified with HM ∗ ( Y, s ). The gen-eral gluing theorem in [7, Equation 3 . c structures, is then m ( u | X, s X ) = (cid:104) ψ ( u | X , s ) , ψ ( u | X , s ) (cid:105) , where the angular brackets denote the natural pairing HM ∗ ( Y, s ) × HM ∗ ( Y, s ) → Z . In our assumptions, the group HM ∗ ( Y, s ) vanishes, so this pairing is zero,and the result follows. (cid:3) IAN AGOL AND FRANCESCO LIN
Y X ˜ XY ¯ YY Y ←− Figure 1.
A double cover of X contains a separating L -space. Remark 2.4.
In fact, for our purposes of understanding gluing formula forSeiberg-Witten invariants, it suffices to consider the reduced invariants withrational coefficients HM ∗ ( Y, s ; Q ). In particular, the previous discussiononly relies on the vanishing of this group. Furthermore, via the universalcoefficients theorem, this is implied by the vanishing of HM ∗ ( Y, s ; Z / Z ), sothat our main result actually applies for the reduced Floer homology groupwith Z / Z -coefficients.Our examples will be based on the following. Corollary 2.5.
Suppose X is a -manifold with b +2 ≥ which admits anembedded non-separating L -space Y . Then X admits infinitely many coverswhich have all vanishing Seiberg-Witten invariants.Proof. Consider the double cover ˜ X of X formed by gluing together twocopies W and W of the cobordism from Y to Y obtained by cutting X along Y , see Figure 1. Consider a properly embedded path γ ⊂ W betweenthe two copies of Y , and denote by T its tubular neighborhood. We thenhave the decomposition X = ( W \ T ) ∪ ( W \ T ), where the two manifolds areglued along a copy of Y Y ; here Y denotes Y with the opposite orientation.The latter is an L -space [10, Section 4], and both W \ T and W \ T have b +2 ≥
1, so we conclude. Of course, we can modify this construction toprovide infinitely many examples. (cid:3) Geodesic hypersurfaces in arithmetic hyperbolic 4-manifolds
In this section, we will discuss various properties of arithmetic hyper-bolic lattices. For the general case of arithmetic lattices, see [20], and forthe 3-dimensional case, consult [13]. We first review the definitions andconstruction of arithmetic manifolds of simplest type.
YPERBOLIC FOUR-MANIFOLDS WITH VANISHING SEIBERG-WITTEN INVARIANTS5
Definition 3.1.
Let G be a group, H , H ≤ G be subgroups. We say that H is commensurable in G with H if [ H : H ∩ H ] < ∞ , [ H : H ∩ H ] < ∞ . Definition 3.2.
Consider a non-degenerate quadratic form q : k n +1 → k fora totally real number field k ⊂ R with ring of integers O k . Assume that q isLorentzian, i.e. has signature ( n,
1) over R . Moreover, for each non-trivialembedding σ : k → R , assume that σ ◦ q is positive definite. Let O ( q ; k )denote the group of matrices preserving k , i.e. linear transforms A : k n +1 → k n +1 such that q ◦ A = q . Then the subgroup O ( q ; O k ) ⊂ O ( q ; k ) ⊂ O ( q ; R )is a lattice, and acts discretely on the hyperboloid of two sheets H = { x ∈ R n +1 | q ( x ) = − } . Up to isometry, the group O ( q ; R ) ∼ = O ( n, R ), theorthogonal group associated to the quadratic form − x + x + · · · + x n .Projectivizing, P O ( q ; O k ) acts discretely on hyperbolic space H n , which isthe quotient of the hyperboloid H by the antipodal map. A hyperbolicorbifold H n / Γ is said to be of simplest type if Γ is commensurable (up toconjugacy) with
P O ( q ; O k ) for some such q .Example: Let q n : k n +1 → k be defined by q n ( x , x , . . . , x n ) = −√ x + x + · · · + x n over the field k = Q ( √ σ : k → k be the Galoisautomorphism induced by σ ( √
2) = −√
2. Then σ ◦ q n ( x , . . . , x n ) = √ x + x + · · · + x n is positive definite. Hence P O ( q n ; Z [ √ H n . See [20, § Definition 3.3.
Let G be a group. Then G (2) = (cid:104) g | g ∈ G (cid:105) .If G is finitely generated, then G (2) is finite-index in G , and G/G (2) is anelementary abelian 2-group.
Theorem 3.4.
Let M be an orientable hyperbolic arithmetic 3-manifold ofsimplest type with H ( M ; Z /
2) = 0 and not defined over Q . Then M embedsas a totally geodesic non-separating submanifold in a compact arithmetichyperbolic 4-manifold.Proof. Let Γ = π ( M ) ≤ Isom + ( H ). Since M is a Z / Z -homology sphere,Γ (2) = Γ. By [6, Theorem 1.1 (2)], H n / Γ (2) ∼ = M embeds as a totally geodesicsubmanifold of a closed orientable hyperbolic 4-manifold W (the fact that M is not defined over Q implies that W is compact). Briefly, this is proved byshowing that Γ (2) ≤ P O ( q ; k ) so that it is commensurable with P O ( q ; O k )for some Lorentzian quadratic form q : k → k . Taking the quadratic form Q d = dy + q, d ∈ N , we get an embedding of P O ( q ; O k ) < P O ( Q d ; O k )
In this picture, the numbers indicate the branch-ing. The top picture has an obvious order 2 rotational sym-metry along the axis depicted by the big dot. The quotientis the link in S depicted on the bottom left. This is isotopicto the link on the right (which is topologically the same, butwith different branchings). Now, the curve with branching 2is the 3-braid σ σ − , so that taking the n -fold branched coveralong the other component we see that M n is the brancheddouble cover over ( σ σ − ) n .4. Examples
The
Fibonacci manifold M n is the cyclic branched n -fold cover over thefigure-eight knot. For n = 2 we obtain a lens space, for n = 3 the Hantzche-Wendt manifold, while for n ≥ n the Fibonacci manifold M n is an L -space. To see this, recallfrom [19] that M n is the branched double cover over the closure of the 3-braid ( σ σ − ) n (see Figure 2), which is alternating. Using the surgery exacttriangle [8], these can be shown to be L -spaces as in the context of HeegaardFloer homology [15], with the caveat that in our setting the computationonly holds with coefficients in Z / Z ; on the other hand this is enough forour purposes, see Remark 2.4. Notice also that for n (cid:54) = 0 modulo 3, theclosure is a knot, so that M n is a Z / Z -homology sphere.By [5], M n is arithmetic when n = 4 , , , ,
12. Of these examples, n =4 , , Z / Z homology spheres. The only one of these three which is sim-plest type and not defined over Q is M . This is example [13, 13.7.4(a)(iii)],which has invariant trace field a quartic field. As they point out, this is com-mensurable with a tetrahedral group [13, 13.7.4(a)(i)] which is simplest type YPERBOLIC FOUR-MANIFOLDS WITH VANISHING SEIBERG-WITTEN INVARIANTS7 and not defined over Q by [12, Theorem 1]. It is defined over a quadraticform over the field Q ( √ M has a non-separating embedding into a closedorientable hyperbolic 4-manifold W . We may assume that χ ( W ) > b +2 ( W ) >
1. Thus by Corollary2.5, these embed into a hyperbolic 4-manifold with vanishing Seiberg-Witteninvariants. This completes the proof of Theorem 1.1.
Remark 4.1.
One may also get other examples by cutting and doublingor using the interbreeding technique of Gromov-Piatetskii-Shapiro to getnon-arithmetic examples. One can isometrically embed this L -space M ininfinitely many incommensurable hyperbolic 4-manifolds via the method of[6] by taking the forms Q and Q d in the proof of Theorem 3.4 so that d is square-free in k = Q ( √ M n as a non-separating hypersurface [4, § Conclusion
We conclude by pointing out some natural questions related to our method.(1) Can one find an explicit hyperbolic example (such as the Davis man-ifold or the manifolds described in [11]) that satisfies the propertiesof Proposition 2.2? Recall that the Davis manifold has b = 24 and b +2 = 36 [16], so that all moduli spaces have odd dimension.(2) Can one embed any orientable hyperbolic 3-manifold of simple typeas a geodesic hypersurface in an orientable hyperbolic 4-manifold?More generally, can one show that orientable hyperbolic 3-manifoldshave quasiconvex embeddings into orientable hyperbolic 4-manifolds?(3) Can one use bordered Floer theory to compute the Seiberg-Witteninvariants of Haken hyperbolic 4-manifolds (in the sense of [3])?(4) Which commensurability classes of compact hyperbolic 3-manifold ofthe simplest type contain L -spaces? Note that it is not even knownif there are infinitely many commensurability classes of arithmeticrational homology 3-spheres. References [1] Nicolas Bergeron,
Premier nombre de Betti et spectre du laplacien de certainesvari´et´es hyperboliques , Enseign. Math. (2) (2000), no. 1-2, 109–137.[2] Chern, Shiing-Shen, On curvature and characteristic classes of a Riemann manifold ,Abh. Math. Sem. Univ. Hamburg 20 (1955), 117-126.[3] Bell Foozwell and Hyam Rubinstein,
Introduction to the theory of Haken n -manifolds ,Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math.Soc., Providence, RI, 2011, pp. 71–84.[4] Gromov, M. and Piatetski-Shapiro, I. Nonarithmetic groups in Lobachevsky spaces.
Inst. des Hautes ´Etudes Sci. Publ. Math. No. 66 (1988), 93-103.
IAN AGOL AND FRANCESCO LIN [5] Hugh M. Hilden, Mar´ıa Teresa Lozano, and Jos´e Mar´ıa Montesinos-Amilibia,
Thearithmeticity of the figure eight knot orbifolds , Topology ’90 (Columbus, OH, 1990)169–183.[6] Alexander Kolpakov, Alan W. Reid, and Leone Slavich,
Embedding arithmetic hy-perbolic manifolds , Mathematical Research Letters (2018), no. 4, 1305–1328,arXiv:1703.10561.[7] Kronheimer, Peter and Mrowka, Tomasz, Monopoles and three-manifolds , New Math-ematical Monographs, 10. Cambridge University Press, Cambridge, 2007. xii+796 pp[8] Kronheimer, P. and Mrowka, T. and Ozsv´ath, P. and Szab´o, Z.,
Monopoles and lensspace surgeries , Ann. of Math. (2) 165 (2007), no. 2, 457-546.[9] Claude LeBrun,
Hyperbolic manifolds, harmonic forms, and Seiberg-Witten invari-ants , Proceedings of the Euroconference on Partial Differential Equations and theirApplications to Geometry and Physics (Castelvecchio Pascoli, 2000), vol. 91, 2002,pp. 137–154.[10] Lin, Francesco, Pin(2) -monopole Floer homology, higher compositions and connectedsums , J. Topol. 10 (2017), no. 4, 921-969.[11] C. Long,
Small volume closed hyperbolic 4-manifolds , Bull. Lond. Math. Soc. (2008), no. 5, 913–916.[12] C. Maclachlan and A. W. Reid, The arithmetic structure of tetrahedral groups ofhyperbolic isometries , Mathematika (1989), no. 2, 221–240 (1990).[13] Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds , Grad-uate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003.[14] Marston Conder and Colin Maclachlan,
Compact hyperbolic 4-manifolds of small vol-ume , Proc. Amer. Math. Soc. (2005), no. 8, 2469–2476.[15] Ozsv´ath, Peter and Szab´o, Zolt´an,
On the Heegaard Floer homology of brancheddouble-covers , Adv. Math. 194 (2005), no. 1, 1-33.[16] Ratcliffe, John G. and Tschantz, Steven T.,
On the Davis hyperbolic 4-manifold ,Topology Appl. 111 (2001), no. 3, 327-342.[17] Alan W. Reid,
Arithmetic kleinian graups and their fuchsian subgroups , Ph.D. thesis,Aberdeen, 1987.[18] Alan W. Reid,
Surface subgroups of mapping class groups , Problems on mapping classgroups and related topics, 257-268, Proc. Sympos. Pure Math., 74, Amer. Math. Soc.,Providence, RI, 2006.[19] Vesnin, A. Yu. and Mednykh, A. D.,
Fibonacci manifolds as two-sheeted coverings overa three-dimensional sphere, and the Meyerhoff-Neumann conjecture , Siberian Math.J. 37 (1996), no. 3, 461-467[20] Dave Witte Morris,
Introduction to arithmetic groups , Deductive Press,http://deductivepress.ca/, 2015.
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