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On the Handlebody Decomposition Associated to a Lefschetz Fibration

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On the Handlebody Decomposition Associated to a Lefschetz Fibration PACIFIC JOURNAL OF MATHEMATICS Vol. 89, No. 1, 1980 ON THE HANDLEBODY DECOMPOSITION ASSOCIATED TO A LEFSCHETZ FIBRATION A. KAS The classical Lef schetz hyperplane theorem in algebraic geometry describes the homology of a protective algebraic manifold M in terms of "simpler" data, namely the homology of a hyperplane section X of M and the vanishing cycles of a Lef schetz pencil containing X. This paper is a first step in proving a diffeomorphism version of the Lefschetz hyperplane theorem, namely a description of the diffeomor- phism type of M in terms of "simpler" data. Let M be the manifold obtained from M by blowing up the axis of a Lefschetz pencil. There is a holomorphic mapping f:M—>CP* which is a Lefschetz fibration, i.e., / has only nondegenerate critical points (in the complex sense). Using the Morse function z->\f(z)\2 on M—f'H00)* one ob- tains a handlebody decomposition of M—/"H00) which may be described as follows: Let X—f'^a) be a regular fiber of /. Choose a system of smooth arcs γlt , γμ starting at a and ending at the critical values of / such that the γ'β are pairwise disjoint except for their common initial point. The γ'β are ordered such that the tangent vectors fί(0), • •, γμ(0) rotate in a counter clockwise manner. To each γj one may associated a "vanishing cycle9', i.e., an imbedding φr. Sn —> X (dim X—2n) defined up to isotopy, together with a bundle isomorphism φ)\τ->v where τ is the tangent bundle to Sn and v is the normal bundle of Sn in X corresponding to the imbedding φj. φj together with the well known bundle isomorphism rφε^εn+1 determines a trivialization of the normal bundle of e2icijlμxφj(Sn) in S*xX. This trivialization allows one to attach a w-handle to D2xX with the core e2πijlμxφj(Sn). If this is done for each j,j=l, •••,/<, the re- sulting manifold is diffeomorphic to M-(tubular neighbor- hood of /"Hoo)). Using the bundle isomorphism φj and the tubular neigh- borhood theorem one may identify a closed tubular neigh- borhood T of φj(Sn) in X with the tangent unit disk bundle to Sn. One may then define a diffeomorphism, up to isotopy, δji X —• X with support in T. δj is a generalization of the classical Dehn-Lickorish twist, δj is the geometric monodro- my corresponding to the jth critical value of /. It follows that the composition δμ° °o1 is smoothly isotopic to the identity 1X:X-+X. A smooth isotopy is given by a smooth arc / in Diff(X) joining the identity to δμo-- oδλ. The choice of /, up to homotopy, determines the way in which one closes off M-(tubular neighborhood of / H00)) to obtain M. 89 90 A. KAS Thus the diffeomorphism type of M is determined by the invariants φltφί, -",φμ,φμ and {λ}, the homotopy class of L Conversely, given a compact oriented 2n dimensional mani- n fold X, imbeddings φj\S ->X1 j=l, •••, μ, and bundle iso- morphisms φjiτ^vj such that <V'*O(5i is smoothly isotopic to lχ, and a homotopy class {λ\ of arcs in Diff(X) with O(5 initial point lx and end point <V* i, one may construct a 2n+2 dimensional manifold M and a Lefschetz fibration f:M—» CP1. It is shown that in the case n—1, apart from certain exceptions, M is uniquely determined by φlf , φμ> i.e., the bundle isomorphisms φ[, " ,φμ and the smooth iso- topy class {λ} are superfluous. 0. Introduction. The classical Lefschetz hyperplane theorem in algebraic geometry describes the homology groups of an alge- braic manifold M in terms of those of a hyperplane section X and in terms of the "vanishing cycles" of X. This paper was inspired by the Morse theoretic proof of the Lefschetz hyperplane theorem due to Andreotti and Frankel [1]. Their approach is to blow up the base locus of a generic Lefschetz pencil so as to obtain a manifold M and a "Lefschetz fibration" /: M -> P\C). They then use the Morse function |/|2 to describe M, at least up to homotopy type, and finally they show how to relate the homology groups of M to those of M. Now according to Smale's handlebody theory [11], it should be possible to use the Morse function |/|2 to determine the full diffeomorphism class of M, not just its homotopy type. In order to do this we must describe the framings of the imbedded spheres (vanishing cycles) corresponding to the critical points of I/I2. In general, the framing associated to a critical point of index n + 1, has an ambiguity measured by πn(S0(n + 1)). In our situa- tion, we can improve this to πn(S0(ri)). This completely determines the framing in certain cases, most notably if JbΓ is a complex sur- face. In this paper, we describe a set of invariants associated to a Lefschetz fibration /: Λf —> P\ which allows one, in principle, to give a handlebody decompositions of certain complex surfaces. There is a certain amount of overlap between some of the ideas of this paper and certain papers of B. Moishezon and R. Mandelbaum (cf. e.g., [6], [7], [10]). 1* The framings associated with a Lefschetz fibration* Let M be a smooth manifold of dimension ^2 and let f:M—>S2 be a smooth mapping. A point peM will be called a critical point of / 2 if the differential dfp: T(M)P -> T(S )f{p) is not surjective. Now let M be a closed compact oriented smooth manifold of even dimension, ON THE HANDLEBODY DECOMPOSITION 91 say dim M = 2n + 2, n ;> 0, and assume that f: M-> S2 is a surjec- tive mapping with a finite number of critical points. We will identify S2 with the extended complex plane C(J°°. If z€S2 is a regular value of /, then /"^(sO is called a regular fiber of /. It is clear that up to diffeomorphism, the regular fibers of / are inde- pendent of the regular value z. DEFINITION 1.1. The smooth mapping f:M~>S2 will be called a Lefschetz fibration if each critical point p of f admits a coordinate neighborhood with complex valued coordinates (wlf , wΛ+1), con- sistent with the given orientation of M, and if f(p) has a coordinate neighborhood with a complex coordinate z, consistent with the orientation of S2, such that locally, / has the form: f(w) = z0 + w\Λ DEFINITION 1.2. Two Lefschetz fibrations 2 2 UM >S , and f2:M > S are said to be equivalent, if /2 = flr°/i where g is an orientation preserving diffeomorphism of S2. Let X denote any regular fiber of the Lefschetz fibration f:M~->S2. Notice that up to diffeomorphism, X only depends on the equivalence class of the Lefschetz fibration f:M-^S2. Notice also, that X has a unique orientation consistent with the orienta- tions of M and S2. We wish to describe a handlebody decomposition of M associated to the Lefschetz fibration f:M-*S2. We will assume that a hand- lebody decomposition of X is already known. We first recall some standard facts about handles and handle- bodies. Let JV be a manifold with boundary and let n •= dim N. Let Φ:Sk-ιxDn~k >dN k n k be a smooth imbedding. Form the union Nτ= N\JΦD xD - where we identify each point of S1*-1 x Dn-kczd(Dk x Dn~k) with its image under Φ. JVΊ is a manifold with boundary and corner points. Let V denote the unique manifold (possibly with boundary) obtained from Nt by "straightening the corners" of Nλ (cf. [2]). DEFINITION 1.3. V is called: the manifold obtained from N by attaching a &-handle along Φ. It is easy to see that, up to diffeomorphism, V depends only on the smooth isotopy class of Φ. Let Φo denote the restriction of 92 A. KAS Φ to JS*""1 XO. It follows easily from the tubular neighborhood theorem [9], that Φ is determined, up to smooth isotopy, by a bundle isomorphism: where en~k is the trivial n — k bundle on Sh~\ and v is the normal 1 bundle of S*" in dN under the imbedding Φo. Thus the isotopy class of the imbedding Φ is determined by: ( i ) A smooth isotopy class of imbeddings 1 Φo: S"- > dN (ii) For each Φo in (i), a smooth isotopy class of bundle iso- morphisms: Notice that (i) is a "knot invariant". As for (ii), the distinct bundle isomorphisms, up to smooth isotopy, are classified by the r group πk^(S0(n — k)). Φ is called a framing of Φo. Let F:M~>R be a Morse function, and let ceR be a critical value such that F~\c) contains a single critical point p, where F has index λ at p. For each real number aeR, let Ma = {xeM\F(x) ^ a}. Then for ε > 0 sufficiently small, Mc+ε is diffeo- morphic to the manifold obtained from Me-S by attaching a λ-handle λ ι n λ along Φ: S ~ x D ~ —> dMc-ζ. To construct Φ explicitely, one may choose (by Morse's lemma) a system of coordinates x19 ---,xn in M centered at p, such that F(xu , xn) — c — x\ — — x\ + x\+ι + • λ + xl (cf. e.g., [8]). Then if ξ - (ςl9 , ξλ)eS -\ V = (%+i, , V.) e nλ 1 n λ D y Φ: S*- x D ~ - > F~\c - ε) is defined by setting where (El, Similarly, if F~\c) contains several nondegenerate critical points Pi, '' ,pμ of indices λ1? * ,λΛ, then Mc+ε is diffeomorphic to the manifold obtained from Mc-Z by attaching μ handles, where the jth λ ι λ handle is a λrhandle attached along Φά: S '-xD*- *-> F\c — έ), 3 = 1, , μ, and where the images of the Φά are disjoint.
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