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 → CP^{1} 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 γ_{1},⋯,γ_{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 γ_{1}′(0),⋯,γ_{μ}′(0) rotate in a counter clockwise
manner. To each γ_{j} one may associated a “vanishing cycle”, i.e., an imbedding
ϕ_{j} : S^{n} → X (dimX = 2n) defined up to isotopy, together with a bundle isomorphism
ϕ_{j}′ : τ → ν where τ is the tangent bundle to S^{n} and ν is the normal bundle of S^{n} in
X corresponding to the imbedding ϕ_{j}. ϕ_{j}′ together with the well known bundle
isomorphism τ ⊕ 𝜖 ≃ 𝜖^{n+1} determines a trivialization of the normal bundle
of e^{2πij∕μ} × ϕ_{j}(S^{n}) in S^{1} × X. This trivialization allows one to attach a
nhandle to D^{2} × X with the core e^{2πij}∕μ × ϕ_{j}(S^{n}). 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 ϕ_{j}′ and the tubular neighborhood theorem one
may identify a closed tubular neighborhood T of ϕ_{j}(S^{n}) in X with the tangent unit
disk bundle to S^{n}. One may then define a diffeomorphism, up to isotopy, δ_{j} : X → X
with support in T. δ_{j} is a generalization of the classical DehnLickorish twist. δ_{j} is
the geometric monodromy corresponding to the jth critical value of f. It follows
that the composition δ_{μ} ∘⋯∘δ_{1} is smoothly isotopic to the identity 1_{X} : X → X. A
smooth isotopy is given by a smooth arc λ in Diff(X) joining the identity to
δ_{p} ∘⋯ ∘ δ_{1}. The choice of λ, up to homotopy, determines the way in which
one closes off M(tubular neighborhood of f^{1}(∞)) to obtain M. Thus the
diffeomorphism type of M is determined by the invariants ϕ_{1},ϕ_{1}′,⋯,ϕ_{μ},ϕ_{μ}′
and {λ}, the homotopy class of λ. Conversely, given a compact oriented 2n
dimensional manifold X, imbeddings ϕ_{j} : S^{n} → X, j = 1,⋯,μ, and bundle
isomorphisms ϕ_{j}′ : τ ∼ ν_{j} such that δ_{μ} ∘⋯ ∘ δ_{1} is smoothly isotopic to 1_{x},
and a homotopy class {λ} of arcs in Diff(X) with initial point 1_{x} and end
point δ_{p} ∘⋯ ∘ δ_{1}, one may construct a 2n + 2 dimensional manifold M and
a Lefschetz fibration f : M → CP^{1}. It is shown that in the case n = 1,
apart from certain exceptions, M is uniquely determined by ϕ_{1},⋯,ϕ_{μ}, i.e.,
the bundle isomorphisms ϕ_{1}′,⋯,ϕ_{μ}′ and the smooth isotopy class {λ} are
superfluous.
