Singular Lefschetz pencils

Abstract

We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4-manifold equipped with a "near-symplectic" structure (ie, a closed 2-form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4-manifold (X, ω) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S 1 which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2-form. Conversely, from such a decomposition one can recover a near-symplectic structure.

Constructing Lefschetz-type fibrations on four-manifolds

“…We will present our version of this construction in Remark 6.2 at the end of the paper. Note here that the exceptional divisors are not sections of the BLF, and that the round 1-handle singularities do not all project to parallel copies of the equator, so this does not quite recover the main result of [4]. However, this construction may be sufficient as input into Perutz's program to construct smooth invariants from BLFs.…”

“…This theorem can be compared to work of Auroux, Donaldson and Katzarkov [4] and of Etnyre and Fuller [12]. In the first it is shown that if X 4 has a near-symplectic form (which it does when b C 2 > 0), then X 4 is a broken Lefschetz pencil (BLP).…”

“…(We call these "round 1-handle singularities".) Such a fibration was called a "singular LF" in Auroux-Donaldson-Katzarkov [4], and the singularities were called "indefinite quadratic singularities" there. Finally, a broken achiral LF (BALF) is one in which all three types of singularities are allowed.…”

Vortices and a TQFT for Lefschetz fibrations on 4–manifolds

“…This choice of c corresponds to the choice made in the construction of periodic Floer homology; see [15]. In this case, for any other z c 2 H 2 .X I ‫/ޒ‬ evaluating positively on the fibers of f , z ƒ h;z c and ƒ h;z c are obviously algebras over ROEkerhc 1 .s h /; i D z ƒ h;˙c 1 .s h / , so if hc 1 .s h /; fiberi ¤ 0 we have a well defined group HF.Y; F; h;˙c 1 .s h /; I z ƒ h;z c /, where the sign at the front of˙c 1 .s h / is chosen to make its evaluation on the fiber positive.…”

“…In [35], Perutz uses and extends constructions similar to this in order to construct a "Lagrangian matching invariant" for the singular Lefschetz fibrations constructed in Auroux-Donaldson-Katzarkov [1] (which exist on blowups for any 4-manifold with b C > 0, though it is not known whether Perutz's invariant is independent of the choice of singular Lefschetz fibration on a given 4-manifold), and conjectures that this new invariant, too, agrees with the Seiberg-Witten invariant. [17] to be the homology of a chain complex CP .…”