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On page 6 of the article Symplectic Lefschetz fibrations with arbitrary fundamental groups, the authors state that for a Lefschetz fibration (with total space of dimension 4) the retraction of the regular fibre onto a critical fibre contains a vanishing cycle.

  • Why does such a vanishing cycle exist?
  • Is existence of vanishing cycles guaranteed when the total space has dimension higher than 4?
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1 Answer 1

Vanishing cycles always exist iff you have at least one critical point $x$ in your Lefschetz fibration f ; in this case there are Complex charts $(w,z)$ with $f(w,z)= w^2+z^2$ . Now, this map has a critical value at $w^2+ z^2 =0 $ You get this critical value by collapsing maps $w^2+z^2= \epsilon$ ; $\epsilon \rightarrow 0$

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