Vol. 89, No. 1, 1980

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On the handlebody decomposition associated to a Lefschetz fibration

Arnold Samuel Kas

Vol. 89 (1980), No. 1, 89–104

The classical Lefschetz 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 Lefschetz pencil containing X. This paper is a first step in proving a diffeomorphism version of the Lefschetz hyperplane theorem, namely a description of the diffeomorphism 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 CP1 which is a Lefschetz fibration, i.e., f has only nondegenerate critical points (in the complex sense). Using the Morse function z →|f(z)|2 on M f1(), one obtains a handlebody decomposition of M f1() which may be described as follows: Let X = f1(a) be a regular fiber of f. Choose a system of smooth arcs γ1,μ starting at a and ending at the critical values of f such that the γ’s are pairwise disjoint except for their common initial point. The γ’s are ordered such that the tangent vectors γ1(0),μ(0) rotate in a counter clockwise manner. To each γj one may associated a “vanishing cycle”, i.e., an imbedding ϕj : Sn X (dimX = 2n) defined up to isotopy, together with a bundle isomorphism ϕj: τ ν where τ is the tangent bundle to Sn and ν is the normal bundle of Sn in X corresponding to the imbedding ϕj. ϕjtogether with the well known bundle isomorphism τ 𝜖 𝜖n+1 determines a trivialization of the normal bundle of e2πij∕μ × ϕj(Sn) in S1 × X. This trivialization allows one to attach a n-handle to D2 × X with the core e2πij∕μ × ϕj(Sn). If this is done for each j, j = 1,, the resulting manifold is diffeomorphic to M-(tubular neighborhood of f1()).

Using the bundle isomorphism ϕjand the tubular neighborhood theorem one may identify a closed tubular neighborhood T of ϕj(Sn) in X with the tangent unit disk bundle to Sn. One may then define a diffeomorphism, up to isotopy, δj : X X with support in T. δj is a generalization of the classical Dehn-Lickorish twist. δj is the geometric monodromy corresponding to the j-th critical value of f. It follows that the composition δμ δ1 is smoothly isotopic to the identity 1X : X X. A smooth isotopy is given by a smooth arc λ in Diff(X) joining the identity to δp δ1. The choice of λ, up to homotopy, determines the way in which one closes off M-(tubular neighborhood of f1()) to obtain M. Thus the diffeomorphism type of M is determined by the invariants ϕ11,μμ and {λ}, the homotopy class of λ. Conversely, given a compact oriented 2n dimensional manifold X, imbeddings ϕj : Sn X, j = 1,, and bundle isomorphisms ϕj: τ νj such that δμ δ1 is smoothly isotopic to 1x, and a homotopy class {λ} of arcs in Diff(X) with initial point 1x and end point δp δ1, 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 ϕ1,μ, i.e., the bundle isomorphisms ϕ1,μand the smooth isotopy class {λ} are superfluous.

Mathematical Subject Classification 2000
Primary: 57N15
Secondary: 14F35, 14J99
Received: 1 November 1978
Published: 1 July 1980
Arnold Samuel Kas