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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 − f−1(∞), one obtains a
handlebody decomposition of M − f−1(∞) which may be described as follows: Let
X = f−1(a) be a regular fiber of f. Choose a system of smooth arcs γ1,⋯,γμ
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),⋯,γμ′(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. ϕj′ together 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,⋯,μ, the resulting manifold is diffeomorphic to M-(tubular neighborhood of
f−1(∞)).
Using the bundle isomorphism ϕj′ and 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 ϕ1,ϕ1′,⋯,ϕμ,ϕμ′
and {λ}, the homotopy class of λ. Conversely, given a compact oriented 2n
dimensional manifold X, imbeddings ϕj : Sn → X, j = 1,⋯,μ, 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,⋯,ϕμ, i.e.,
the bundle isomorphisms ϕ1′,⋯,ϕμ′ and the smooth isotopy class {λ} are
superfluous.
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