# Four Manifolds: Confluence of High and Low Dimensions

**Location: ** Fields Institute, Stewart Library

## Description

This workshop will bring together researchers from the worlds of low- and high-dimensional manifold topology, especially those with interests in the study of 4-manifolds.

The field of 4-dimensional manifolds is shrouded in mystery. Fundamental open problems abound, among them the smooth Poincaré conjecture, the smooth Schoenflies problem, and the topological surgery exact sequence for free groups. Indeed, there is not a single smoothable topological $4$-manifold whose smooth structures are classified. Questions about the existence and uniqueness of families of diffeomorphisms for any given smooth $4$-manifold remain equally intractable.

On one hand, topological $4$-manifolds behave much like high-dimensional manifolds. By the foundational work of Freedman and Quinn, surgery theory and the $s$-cobordism theorem apply in the topological category and can be used to classify compact simply connected topological $4$-manifolds in terms of the intersection form. On the other hand, smooth $4$-manifolds exhibit remarkable behaviour, setting them aside from manifolds in all other dimensions. One of the simplest $4$-manifolds, $\mathbb{R}^4$, has uncountably many distinct smooth structures. Furthermore, Gompf has recently shown that there exist infinitely many pairwise non-isotopic diffeomorphisms of an exotic $\mathbb{R}^4$.

However, up to stabilisation by connected sum with $S^2 \times S^2$, smooth $4$-manifolds can be classified using surgery, as if they were high-dimensional manifolds. Moreover, Gabai's $4$-dimensional light bulb theorem shows that homotopy implies isotopy for spheres in $S^2 \times S^2$ having a geometric dual. This result sparked recent interest knotted surfaces in 4-manifolds.

For high dimensional manifolds, the Galatius-Madsen-Tillman-Weiss determination of the homotopy type of the cobordism category, together with homological stability results of Galatius and Randal-Williams, led to computations of the homology of diffeomorphism groups of manifolds that have been suitably stabilised by copies of $S^k \times S^k$, where $k$ is at least three. Thus notions of stability have appeared in recent advances in quite different guises.

Attempts to understand $4$-manifolds and embedded surfaces therein make use of techniques from a variety of related fields, such as Seiberg-Witten gauge theory, Heegaard-Floer homology, Kirby's calculus of links, symplectic topology, and Morse theory. This conference aims to create interactions between experts in these techniques and other areas of manifold topology such as homological stability phenomena, embedding calculus, surgery theory, field theory, and more.