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1. Lefschetz properties through a topological lens

   https://doi.org/10.2478/aupcsm-2025-0001  

   Seceleanu, Alexandra (June 2025, Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica)

   Abstract These notes were prepared for theLefschetz Preparatory School, a graduate summer course held in Krakow, May 6–10, 2024. They present the story of the algebraic Lefschetz properties from their origin in algebraic geometry to some recent developments in commutative algebra. The common thread of the notes is a bias towards topics surrounding the algebraic Lefschetz properties that have a topological flavor. These range from the Hard Lefschetz Theorem for cohomology rings to commutative algebraic analogues of these rings, namely artinian Gorenstein rings, and topologically motivated operations among such rings. 

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2. On the Frøyshov invariant and monopole Lefschetz number

   https://doi.org/10.4310/jdg/1683307008  

   Lin, Jianfeng; Ruberman, Daniel; Saveliev, Nikolai (March 2023, Journal of Differential Geometry)

   Given an involution on a rational homology 3-sphere Y with quotient the 3-sphere, we prove a formula for the Lefschetz num- ber of the map induced by this involution in the reduced mono- pole Floer homology. This formula is motivated by a variant of Witten’s conjecture relating the Donaldson and Seiberg–Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic ar- gument, making use of an exact triangle in monopole Floer homol- ogy, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants as- sociated to spin structures on Y . We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology. 

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3. Geography of symplectic Lefschetz fibrations and rational blowdowns

   https://doi.org/10.1090/tran/8968  

   Baykur, R; Korkmaz, Mustafa; Simone, Jonathan (July 2024, Transactions of the American Mathematical Society)

   We produce simply-connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides asymplecticextension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blowdown configurations that yield new constructions of small symplectic exotic $4$–manifolds. We present an example of a construction of a minimal symplectic exotic $\mathbb{C}\mathbb{P}{}^2\mathrm{\#<\#comment/>}5\stackrel{¯<\#comment/>}{\mathbb{C}\mathbb{P}}{}^2$through this procedure applied to a genus– $3$fibration. 

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4. Sectorial descent for wrapped Fukaya categories

   https://doi.org/10.1090/jams/1035  

   Ganatra, Sheel; Pardon, John; Shende, Vivek (October 2023, Journal of the American Mathematical Society)

   We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a ‘stop removal equals localization’ result, and (4) that the Fukaya–Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful. 

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5. Lefschetz fibrations with arbitrary signature

   https://doi.org/10.4171/JEMS/1326  

   Baykur, R İnanç; Hamada, Noriyuki (June 2024, Journal of the European Mathematical Society)

   We develop techniques to construct explicit symplectic Lefschetz fibrations over the2-sphere with any prescribed signature\sigmaand any spin type when\sigmais divisible by16. This solves a long-standing conjecture on the existence of such fibrations with positive signature. As applications, we produce symplectic4-manifolds that are homeomorphic but not diffeomorphic to connected sums ofS^2 \times S^2, with the smallest topology known to date, as well as larger examples as symplectic Lefschetz fibrations. 

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