New method for decomposing manifolds into 1-handlebodies.
problem Decomposing manifolds into 1-handlebodies.
method Multisections of surface bundles and fiber bundles over the circle.
result Multisections exist for surface bundles and fiber bundles over the circle.
Multisections generalize Heegaard splittings and trisections to higher dimensions.
problem Decomposing higher-dimensional manifolds into 1-handlebodies with specific intersection properties.
method Considering multisections of higher-dimensional smooth or PL closed orientable manifolds, and associating diagrams to them.
result Multisection diagrams uniquely determine manifolds in all dimensions up to 6.
New proof of Laudenbach and Poénaru's theorem on 4D 1-handlebodies.
problem Proving diffeomorphisms extend to 4D 1-handlebodies and compression bodies.
method Classification of Heegaard splittings and Cerf's theorem on 3-ball diffeomorphisms.
result Every diffeomorphism of the positive boundary extends to the whole compression body.
Efficient multisections found for odd-dimensional tori.
problem Finding optimal multisections of odd-dimensional tori.
method Constructing multisections with specific properties (genus n, symmetry) for odd-dimensional tori. result Optimal multisections of genus n found for odd-dimensional tori. A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the clo…
The paper constructs multisections for m-spun 3-manifolds in higher dimensions.
problem Finding multisections for higher-dimensional manifolds.
method Extending the concept of spun 3-manifolds to higher dimensions, constructing multisections and diagrams.
result Infinitely many examples of non-diffeomorphic multisected manifolds in all dimensions.
Recently Gay and Kirby described a new decomposition of smooth closed 4-manifolds called a trisection. This paper generalises Heegaard splittings of 3-manifolds and trisections of 4-manifolds to all dimensions, using triangulations as a key tool. In particular, we prove that every closed piecewise linear n-mani…
This paper constructs a Weinstein trisection for a surface bundle.
problem Constructing a trisection for a surface bundle.
method Weinstein trisection approach applied to ΣgimesΣh. result Explicit construction of a Weinstein trisection for S2imesS2. New 3D handlebodies in 4-sphere and 5-ball are not isotopic even with same boundary.
problem Existence of non-isotopic handlebodies with identical boundary.
method Construction of specific genus-g handlebodies in S4 and B5. result Proves Budney-Gabai conjecture for genus at least 2.
The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every k≥2, we construct 2k−1 non-diffeomorphic (3k,k)-trisections on infinitely many 4-manifolds. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier's spun trisection…
New modules derived from Khovanov homology for links.
problem Constructing new link homology modules from Khovanov homology.
method Functoriality proof for cobordisms in 4D relative 1-handlebody complements.
result Functoriality of Rozansky-Willis's homology for cobordisms.
New method to compute homology and intersection form of 4-manifolds.
problem Computing homology and intersection form of 4-manifolds.
method Using multisections to define complexes and retraction onto CW-complex.
result Efficient proofs of homology and intersection form from multisection diagrams.
New invariant detects more elements in 4D diffeomorphism group.
problem Detecting more elements in the mapping class group of 4D handlebodies.
method Defined and computed a new invariant (W3)m for π0Diff(aturalmS1imesD3,∂). result Infinitely many non-isotopic separating 3-balls found.
We prove that if M is a CW-complex and ∗ is a 0-cell of M, then the crossed module Π2(M,M1,∗) does not depend on the cellular decomposition of M up to free products with Π2(D2,S1,∗), where M1 is the 1-skeleton of M. From this it follows that if G is a finite crossed module and M is finite, the…
Study framed links in simply connected 4-manifolds, linking to exotic phenomena.
problem Understanding framed links in simply connected 4-manifolds and their relation to exotic phenomena.
method Analyzing the set of framed smoothly slice links on the boundary of a 1-handlebody in a closed, simply connected, smooth 4-manifold.
result Exotic 4-manifolds are distinguished by links that are slice in the exotic manifold but not in the standard D4. We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity Gκ=(∑i<jλi+λj−2κ1)−1, where $λ_1 \leq \hdots \leq λ_n$ denote the curvature eigenvalues and κ is a nonnegative constant. This defines a fully nonlinear parabolic equation, provided t…
The paper studies hypersurfaces in spheres using mean curvature flow with surgery.
problem Studying hypersurfaces in spheres under specific curvature pinching conditions.
method Using mean curvature flow with surgery to preserve and analyze curvature pinching conditions.
result Hypersurfaces satisfying the pinching condition are diffeomorphic to spheres or connected sums of spheres.