Workshop FRUMAM

TUESDAY, NOVEMBER 25, 2025

FRUMAM, second floor seminar room, campus Saint-Charles, Marseille

Golla: Lens spaces in the complex projective plane
Which lens spaces embed smoothly in the complex projective plane, and which collections of lens spaces can be disjointly embedded? Work of Hacking and Prokhorov showed that each solution to the Markov equation gives rise to a triple of lens spaces which embed disjointly, and Evans-Smith showed this accounts for all symplectic embeddings of the standard rational homology balls bounded by these lens spaces. Further embeddings of lens spaces have since been exhibited, including two families of triples which embed disjointly due to Lisca and Parma. I will discuss some new obstructions and some new constructions of triples. This is joint work with Brendan Owens.

Rechtman: When is the geodesic flow of a sphere of revolution left-handed?
On a 3-dimensional sphere, a flow is left-handed if any pair of (ergodic) invariant measures links negatively. Here the linking is defined by taking asymptotic orbits. This property was introduced by É. Ghys in 2009, who proved that in a left-handed flow every collection of periodic orbits is the boundary of a Birkhoff section. A natural question is when the geodesic flow of the two sphere is left-handed.
A. Florio and U. Hryniewicz found bounds on the pinching of curvature that guarantee that the flow is left-handed, but these bounds do not seem to be optimal. I will explain how to find the right bounds in the case of a revolution ellipsoid and more generally for spheres of revolution. This involves understanding how to calculate the number of entanglements of two orbits of the geodesic flow from the corresponding closed geodesics on the 2-sphere. This is joint work with P. Dehornoy.

Meier: Indecomposable Klein bottles with order-4 meridians
A foundational theorem in knot theory states that every knot in 3-space can be expressed uniquely as a connected sum of non-trivial prime knots. To this day, very little is known about the extent to which a similar result might hold for embeddings of closed surfaces in 4-space. For example, it is not known whether the unknotted 2-sphere admits a non-trivial connected sum decomposition, and it is conjectured that every knotted projective plane is the connected sum of a knotted 2-sphere and an unknotted projective plane. In this talk, I’ll survey that pathologies that arise concerning notions of ‘decomposability’ or ‘irreducibility’ for surface-knots in 4-space, and I’ll present an infinite family of knotted Klein bottles that are indecomposable and have order-4 meridians.

Dissler: Decompositions of orientable, compact manifolds into 1–handlebodies
In this presentation, we study decompositions of smooth compact manifolds into 1—handlebodies.  These decompositions are higher-dimensional generalizations of Heegaard splittings of closed 3–manifolds (introduced by Heegaard in 1898) and sutured Heegaard splittings of compact 3–manifolds with boundary (Goda, 1992). In 2016, Gay and Kirby defined trisections of closed 4–manifolds, and relative trisections of 4–manifolds with boundary, as natural extensions of such decompositions. Then, in 2023, Ben Aribi, Courte, Golla and Moussard’s notion of multisections of closed manifolds generalized trisections to higher dimensions.  We show how multisections can be adapted to manifolds with boundary, introducing relative multisections, which generalize relative trisections; however, this construction induces a strong restriction on the boundary, whose connected components have to be either Sn or connected sums of copies of S1 × S(n-1). A nice feature of (relative) multisections is that they are determined by their tridimensional spine. In particular, this enables a diagrammatic representation of (relative) multisections by diagrams. Finally, we quickly introduce a possible generalization of  relative multisections, pseudo-multisections, which alleviates the restriction on the boundary.