Monodromy invariants - From symplectic to smooth manifolds

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Recently, together with Auroux and Donaldson, we have introduced some new invariants of four-dimensional symplectic manifolds. Building on the Moishezon-Teicher braid factorization techniques, we show how to compute fundamental groups of compliments to a ramification curve of generic projection. We also show that these fundamental groups are only homology invariants and outline the computations in some examples. Demonstrating the ubiquity of algebra, we go further and, using Braid factorization, we compute invariants of a derived category of representations of the quiver associated with the Fukaya-Seidel category of the vanishing cycles of a Lefschetz pencil and a structure of a symplectic four-dimensional manifold. This idea is suggested by the homotogical mirror symmetry conjecture of Kontsevich. We do not use it in our computations, although everything is explicit. We outline a procedure for finding homeomorphic, nonsymplectomorphic, four-dimensional symplectic manifolds with the same Saiberg-Witten invariants. This procedure defines invariants in the smooth category as well.

Original languageEnglish (US)
Pages (from-to)85-103
Number of pages19
JournalActa Applicandae Mathematicae
Issue number1-3
StatePublished - Jan 2003
Externally publishedYes


  • Derived categories
  • Lefschetz pencils
  • Moishezon-Teicher braid factorization techniques
  • Symplectic invariants

ASJC Scopus subject areas

  • Applied Mathematics


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