Non-positive Stein-fillable open books of genus one

Vitalijs Brejevs (University of Glasgow)

Monday 18th October, 2021 16:00-17:00 Online


Contact 3-manifolds arise organically as boundaries of symplectic 4-manifolds, so a natural question is the following: Given a contact 3-manifold $ Y $, does there exist a symplectic 4-manifold $ X $, called a filling, such that $ Y $ is the boundary of $ X $ in a way where the contact structure is compatible with the symplectic structure? Stein fillability is one such notion of compatibility that can be explored via open books: decompositions of contact manifolds into surfaces together with self-diffeomorphisms of said surfaces, called monodromies. In this talk I will discuss joint work with Andy Wand in which we exhibit first known Stein-fillable contact manifolds whose open books of genus one have non-positive monodromies. Together with work of Wand and Wendl, this settles the question of correspondence between Stein fillings and positive monodromies for open books of all genera. Our methods rely on a combination of results of J. Conway, Lecuona and Lisca along with some observations about lantern relations in the mapping class group of the two-holed torus.

The talk will be preceded by a tea time at 3:45pm. The Zoom link for the seminar is and the passcode is the genus of the two-dimensional sphere (4 letters, all lowercase).

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