Related papers: Stabilisation, bordism and embedded spheres in 4--…

A strategy for constructing an embedded sphere in a 4-manifold realizing a given homology class which has been successfully applied in the past is to represent the class as a first step stably by an embedded sphere, i.e. after adding…

We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that…

Four observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere $S$ inside $\mathbb{R} P^2\times S^2$ such that performing a Gluck twist on $S$ produces a manifold $Y$ that is…

We say that a topologically embedded 3-sphere in a smoothing of Euclidean 4-space is a barrier provided, roughly, no diffeomorphism of the 4-manifold moves the 3-sphere off itself. In this paper we construct infinitely many one parameter…

In this paper, we collect various structural results to determine when an integral homology $3$--sphere bounds an acyclic smooth $4$--manifold, and when this can be upgraded to a Stein manifold. In a different direction we study whether…

Under certain homological hypotheses on a compact 4-manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. The main examples are the class of finite connected sums of 4-manifolds with…

The main result is that an s-cobordism (topological or smooth) of 4-manifolds has a product structure outside a ``core'' sub s-cobordism. These cores are arranged to have quite a bit of structure, for example they are smooth and abstractly…

We show that the number of double points of smoothly immersed 2-spheres representing certain homology classes of an oriented, smooth, closed, simply-connected 4-manifold X must increase with the complexity of corresponding h-cobordisms from…

We introduce a new stable range invariant for the classification of closed, oriented topological $4$-manifolds (up to $s$-cobordism), after stabilization by connected sum with a uniformly bounded number of copies of $S^2\times S^2$.

We prove that there are homology three-spheres that bound definite four-manifolds, but any such bounding four-manifold must be built out of many handles. The argument uses the homology cobordism invariant $\Gamma$ from instanton Floer…

Let $\pi$ be a group satisfying the Farrell-Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincar\'e duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$…

Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every…

A generalized-homology bordism-theory is constructed, such that for certain manifold homotopy stratified sets (MHSS; Quinn-spaces) homeomorphism-invariant geometric fundamental-classes exist. The construction combines three ideas: Firstly,…

M. Freedman showed that every homology 3-sphere embeds as a locally flat submanifold of $S^4$. This is in striking contrast to the state of our knowledge of smooth embeddings of homology spheres. This book surveys what is presently known…

We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature…

We extend the theory of relative trisections of smooth, compact, oriented $4$-manifolds with connected boundary given by Gay and Kirby to include $4$-manifolds with an arbitrary number of boundary components. Additionally, we provide…

In this note we study whether specific elements in the second homology of specific simply connected closed $4$-manifolds can be represented by smooth or topologically flat embedded spheres.

We investigate the realisability of the Casson-Sullivan invariant for homeomorphisms of smooth $4$-manifolds, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the third cohomology of…

In this article, we answer two questions of Buchanan-McKean (arXiv:2312.08209) about bordism for manifolds with spin$^h$ structures: we establish a Smith isomorphism between the reduced spin$^h$ bordism of $\mathbb{RP}^\infty$ and…

The homology groups of the automorphism group of a free group are known to stabilize as the number of generators of the free group goes to infinity, and this paper relativizes this result to a family of groups that can be defined in terms…