A note on a topological approach to the μ -constant problem in dimension 2
aa r X i v : . [ m a t h . AG ] A p r A NOTE ON A TOPOLOGICAL APPROACH TO THE µ -CONSTANT PROBLEM IN DIMENSION 2 MACIEJ BORODZIK AND STEFAN FRIEDL
Abstract.
We provide an example, which shows that studying homological andhomotopical properties of cobordisms between arbitrary, that is not necessarilynegative, graph manifolds is not enough to prove the µ -constant conjecture of LˆeD˜ung Tr´ang in complex dimension 2. Introduction
Background.
The study of equisingularity is one of the main questions in sin-gularity theory. The systematic study dates back to Zariski [Za65a, Za65b, Za68].One of the milestones is the fact that if the Milnor number µ is constant under thedeformation of isolated hypersurface singularities in C n +1 , and if n = 2, then thedeformation is topologically trivial. This fact was proved by Lˆe, and Lˆe–Ramanujamin the series of papers [Le71, Le72, LR76]. The case n = 1 is simple and relies on afull classification of the singularities of plane curves (compare [Za65a]), while the case n > h -cobordism theorem and the fact, that links of isolated hypersurfacesingularities in C n +1 for n > n > n = 2 has remained open for 40years.There were attempts to solving the µ -constant conjecture using a topological ap-proach, i.e. studying the cobordism of links of singularities. Perron and Shalen in[PS99] proved the µ -constant conjecture under an additional hypothesis on funda-mental groups of the links. They use a detailed study of graph manifolds and a deepunderstanding of the cobordism between graph manifold. A natural question thatarises is: Can one prove the µ -constant conjecture in dimension using only properties ofcobordisms of graph manifolds? A precise formulation of the above question is given in Question 1.4. In this notewe show, that if one admits graph manifolds that are not negative (that is, are not
Date : July 1, 2018.2010
Mathematics Subject Classification. primary: 32S15; secondary: 57M25, 57R65, 57R80.
Key words and phrases. µ -constant problem, graph manifold, deformation of singular points,Milnor number, cobordism of manifolds.The first author is supported by Polish OPUS grant No 2012/05/B/ST1/03195. boundaries of negative definite plumbed manifolds), then the answer to Question 1.4is negative. As links of singularities are negative definite, the counterexample that wegive, does not imply that topological arguments alone are insufficient to prove the µ -constant conjecture in dimension 2. However, it indicates that topological approachesmust take into account negative definiteness of graph manifolds.1.2. The µ -constant problem. We begin with the following formulation of the µ -constant problem. We refer to [Te76, GLS06] for background material on deformationsand equisingularity questions. Question 1.1 (The µ -constant problem) . Suppose we are given a family of complexpolynomial functions F t : ( C n +1 , → ( C , smoothly depending on a parameter t ∈ D ⊂ C , where D is a unit disk. Assume that for each t , the hypersurface X t = F − t (0) has an isolated hypersurface singularity at ∈ C n +1 . Let µ t be the Milnor numberof the singularity of X t at . If µ t is a constant function of t , does it imply that thetopological type of the singularity of X t at does not depend on t ? The results [Le71, Le72, LR76] can be resumed as follows.
Theorem 1.2.
Question 1.1 has an affirmative answer if n = 1 or n ≥ . A possible approach to the problem, and actually the one that is sufficient for cases n = 2 is the following. Let B ⊂ C n +1 be a small closed ball around 0, such that M := X ∩ ∂B is the link of singularity ( X , t ∈ C sufficiently smallso that X t ∩ ∂B is isotopic to M . Let us now choose a smaller ball B t ⊂ C n +1 such M t := X t ∩ ∂B t is the link of the singularity ( X t , W = X t ∩ ( B \ B t ). Then W is a smooth manifold of real dimension 2 n with boundary M ⊔ − M t .The cobordism W has various topological properties which we now summarize inthe following proposition. Proposition 1.3.
The manifolds ( W, M , M t ) satisfy the following properties. (W0) dim R W = 2 n , dim R M = dim R M t = 2 n − , furthermore W, M and M t arecompact and oriented. (W1) If n > , then π ( M t ) = π ( M ) = { e } , if n = 2 , then the image of π ( M t ) in π ( W ) normally generates π ( W ) . (W2) W can be built from M t × [0 , by adding handles of indices , , . . . , n . (W3) If n = 2 , then the manifolds M and M t are oriented, irreducible, graph man-ifolds.Furthermore, if we have the equality of Milnor numbers µ t = µ , then the followingadditional fact is satisfied (W4) The maps H ∗ ( M ; Z ) → H ∗ ( W ; Z ) and H ∗ ( M t ; Z ) → H ∗ ( W ; Z ) induced byinclusions are isomorphisms.Remark. Proposition 1.3 is well known to the experts, for a convenience of the readerwe sketch the proofs or give references.
OPOLOGICAL µ -CONSTANT PROBLEM 3 (W0) is obvious. For n >
2, (W1) is [Mi68, Theorem 6.4]. (W1) for n = 2 and (W4)in the general case can be found in [LR76, proof of Theorem 2.1]. The main idea isto consider the Milnor fibers F t and F for the singularities ( X t ,
0) and ( X , F t has the homotopy type of a wedge of µ t spheres S n , and F has the homotopy type of a wedge of µ spheres S n . Using the equivalence of theMilnor fibration over circle and over a disk (see e.g. [Ham71, Satz 1.5]) we infer that F t ∪ M t W is homeomorphic F . Now if n = 2, then F t and F are simply connected,hence we get (W1) by the van Kampen theorem.If µ t = µ , then F t and F have the same homotopy type. Since the homologygroups of F t and F are zero in all dimensions but 0 and n , the standard homologicalarguments yield (W4).The property (W2) is proved in [AnF59]. Finally, (W3) follows from [Ne81].As it was written in [LR76], in case n >
2, the conditions (W0), (W1) and (W4)imply that W is an h -cobordism and since dim R W ≥ h -cobordism theorem of Smale (see [Sm62, Mi65]) which shows that W is in fact aproduct, which in turn implies that the singularities ( X t ,
0) and ( X ,
0) are topo-logically equivalent. In case n = 2, neither of the manifolds M t and M is simplyconnected, nor does the Whitney trick work (compare [GS99, Section 9.2]). However,since the graph 3-manifolds are somehow rigid, it is still natural, though, to ask thefollowing question. Question 1.4 (Topological µ -constant problem) . Let ( W, M t , M ) satisfy conditions ( W0 ) – ( W4 ) for n = 2 . Does it imply that M and M t are homeomorphic? As is pointed out in [PS99, p. 3], the result of Levine [Lv70, Theorem 3] implies,that if M and M t are homeomorphic, then the singular points ( X t ,
0) and ( X ,
0) aretopologically equivalent. The key element of this observation is the fact that M and M t are simple knots by [Mi68, Lemma 6.4] and the fact, see e.g. [Sae00, Corollary1.3], that M and M t have equivalent Seifert matrices. In [PS99, Proposition 0.5] (seealso [AsF11, p. 1180]) the following theorem was proved. Proposition 1.5.
Question 1.4 has an affirmative answer if we additionally assumethat π ( M t ) surjects onto π ( W ) . The main goal of this note, and actually the content of next section is the followingresult.
Theorem 1.6.
Question 1.4 has a negative answer. A negative answer to Question 1.4
The construction.
Let K ⊂ S be a non-trivial torus knot T ( p, q ). We consider L = K − K , where − K is the mirror image of K with the opposite orientation.Then it is well-known (see e.g. [GS99, p. 210-213]) that L bounds a ribbon disk D in B . This implies that there exists an open ball B ′ ⊂ B with the same center and MACIEJ BORODZIK AND STEFAN FRIEDL smaller radius, such that ∂B ′ ∩ D is an unknot, the distance function on D ∩ ( B \ B ′ )is Morse (for this we might need to move slightly the common center of the two balls)and has only critical points of index 0 and 1, and D ∩ ( B \ B ′ ) is an annulus.Let νD be an open tubular neighbourhood of D . We define X = B \ ( B ′ ∪ νD ).Let Y = ∂X ∩ νD . We define now W = X ∪ Y − X, i.e. we take a double of X along Y . Lemma 2.1.
The boundary of W is a disjoint union of S × S and the double of S \ νL .Proof. We write S ′ = ∂B ′ and J := D ∩ S ′ . Note that J is the unknot. It followsimmediately from the definitions that ∂W is the disjoint union of the double of S \ νL and the double of S ′ \ νJ . The knot J ⊂ S ′ is the unknot, i.e. S ′ \ νJ is a solid torus.The double of S ′ \ νJ is thus canonically homeomorphic to S × S . (cid:3) We define M t = S × S and M as the double of S \ νL . Since L is non-trivial itis clear that M t and M are non-homeomorphic. In the next section we will see thatthe triple ( W, M t , M ) satisfies conditions (W0) to (W4), which thus gives us a proofof Theorem 1.6.2.2. Proof of (W0)–(W4).
The property (W0) is obvious. It is a straightforwardconsequence of Alexander duality and the Mayer–Vietoris sequence that the maps H ∗ ( M ; Z ) → H ∗ ( W ; Z ) and H ∗ ( M t ; Z ) → H ∗ ( W ; Z ) induced by inclusions are iso-morphisms. This proves that (W4) is satisfied.Let us now show (W2). We use the theory of embedded handle calculus as in[GS99, Section 6.2]. Namely, the function ‘distance from the origin’ on B \ B ′ hasonly critical points of index 0 and 1, when restricted to D ∩ ( B \ B ′ ). It follows from[GS99, Proposition 6.2.1] that X can be built from ∂B ′ \ D by adding only handlesof index 1 and 2. By taking the double we obtain that W is built from S × S byadding only handles of index 1 and 2 as desired.We now turn to the proof of (W1). Let x be a generator of π ( S × S ) which wecan represent by a meridian of the unknot J = D ∩ S ′ . We claim that x normallygenerates π ( W ). We denote by Γ the smallest normal subgroup of π ( W ) whichcontains x . We thus have to show that in fact Γ = π ( W ). First note that themeridian of J is homotopic in X , via meridians of the ribbon disk, to a y meridianof the knot L . It is well-known that a meridian normally generates a knot group.We thus see that Im( π ( S \ νL ) → π ( W )) ⊂ Γ. Note that π ( M ) is generated bythe fundamental groups of the two knot exteriors which are glued together. We nowsee that Im( π ( M ) → π ( W )) ⊂ Γ. It follows from (W2) that W is obtained from M t × [0 ,
1] by adding handles of indices 0 , ,
2. By turning the handle decomposition‘upside-down’ we see that we can obtain W from M × [0 ,
1] by adding handles of
OPOLOGICAL µ -CONSTANT PROBLEM 5 indices 2 , ,
4. This implies in particular that π ( M ) → π ( W ) is surjective. It nowfollows that π ( W ) = Γ as desired.We finally turn to the proof of (W3). It is well-known that S \ νK is a Seifertfibered space. Furthermore, we can obtain S \ νL = S \ νK − K by gluing S \ νK and S \ ν − K along their boundaries to S × Σ where Σ is a pair of pants, i.e. Σis obtained by removing three open disks from S . It now follows that S \ νL is agraph manifold. Finally M is obtained by gluing two graph manifolds along theirboundary, which shows that M is also a graph manifold.2.3. Further properties of ( W, M t , M ) . To conclude, we point out that if thetriple (
W, M t , M ) arises from singularity theory, then the following conditions (W5),(W6) and (W7) are additionally satisfied. The properties (W6) and (W7) are notof topological nature. We do not know, if all the properties (W0)–(W7) together,enforce the cobordism W to be a product.(W5) The manifolds M t and M are negative, in other words there exists a negativedefinite plumbing diagram for each of them, see [Ne81].(W6) The manifold W carries a canonical symplectic form, coming from the K¨ahlerstructure on it. The boundary M is convex, while M t is concave with respectto that form. The contact structures induced on M t and M are Milnor fillable(see [Var80, CNP06]).(W7) The manifold W is Stein. Furthermore, M t is the pseudo-convex part of itsboundary and the manifold M is pseudo-concave. Acknowledgement.
We wish to thank Charles Livingston, Andr´as N´emethi and PiotrPrzytycki for valuable discussions. This project was started while we were bothvisiting Indiana University and we would like to express our gratitude for hospitalityof the Mathematics Department of Indiana University. We are also grateful to theanonymous referee for carefully reading an earlier version of this paper.
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