Université de Nantes - Université Paris-Saclay

NANTES-ORSAY SEMINAR ON
SYMPLECTIC AND CONTACT GEOMETRY

Friday, 20 October 2023

András Stipsicz (Rényi Institute Budapest)

ℤ₂-actions on four-manifolds

Abstract: Exotic smooth structures on four-manifolds are one of the most surprising, and at the same time most fascinating phenomena in low dimensional topology. Existence questions of exotic smooth structures on closed simply connected four-manifolds with definite intersection form are of central importance - as a special case, this problem includes the smooth four-dimensional Poincaré conjecture. We show exotic examples of definite four-manifolds with fundamental group ℤ₂. In another direction, ℤ₂-actions on specific exotic four-manifolds provide embeddings of connected sums of the real projective plane ℝℙ² to the four-sphere S⁴ which are topologically standard, but smoothly not. Smooth exotica are proved using Seiberg-Witten theory - we will hint the use of these invariants in this context.

Vincent Florens (Université de Pau)

Topological invariants of line arrangements

Abstract: A line arrangement is a finite set of lines in the complex projective plane ℂℙ². They are particular cases of plane algebraic curves where the components have all degree 1. We are interested in the influence of the combinatorial data of an arrangement on its embedded topology in ℂℙ². The boundary manifold of an arrangement is the common boundary of a regular neighborhood of the arrangement and its exterior. This is a graph three-manifold, in the sense of Waldhausen, whose topology is completely determined by the combinatorics. We use the inclusion map of the boundary manifold in the exterior to construct a new topological invariant of arrangements. This is a joint work with E. Artal and A. Rodau.

Discussions