Exposes two methods for constructing flat surfaces in 4D spaces.
problem Building locally flat embedded surfaces in 4-manifolds.
method Direct methods and surgery theory.
result Every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus.
We prove a concordance version of the 4-dimensional light bulb theorem for π1-negligible compact orientable surfaces, where there is a framed but not necessarily embedded dual sphere. That is, we show that if F0 and F1 are such surfaces in a 4-manifold X that are homotopic and there exists an immersed framed…
The article examines how many stabilizations are needed to transform 5D s-cobordisms into product cobordisms.
problem Understanding the number of stabilizations required to convert 5D s-cobordisms into product cobordisms.
method Utilizes Gabai's 4D light bulb theorem and Freedman Quinn invariant.
result Determines the number of stabilizations needed for 5D s-cobordisms.
2-spheres in 4-manifolds have complete concordance obstructions if they have immersed dual spheres.
problem Determining concordance of 2-spheres in 4-manifolds.
method Adapting Stong's methods to 4-manifolds, using self-intersection sets and concordance obstructions.
result Complete concordance obstructions for 2-spheres with immersed dual spheres in 4-manifolds.
Homology handles with trivial Alexander polynomial bound a 3D sphere.
problem Understanding when homology handles bound 3D spheres.
method Using Freedman and Quinn's result for Z-homology 3-spheres. result A distinguished homology handle with trivial Alexander polynomial bounds a homology S1imesD3. New surfaces in 3D manifolds are found that cannot be smoothly deformed into each other.
problem Finding surfaces in 3D manifolds that cannot be smoothly deformed into each other.
method Constructing infinite families of homotopic surfaces in closed genus-g surfaces, showing they are not smoothly image-concordant. result Closed surfaces with common framed dual sphere can be π1-injective but not smoothly image-concordant. Modified proof constructs dual spheres for 4-manifolds.
problem Prove dual spheres for 4-manifolds.
method Modified proof of disc embedding theorem, geometric construction.
result Constructs geometrically dual spheres.
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 certain product geometries. Most of these compact manifolds have non-vanishing sec…
Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite con…
New 4-manifold examples show necessary conditions for 4D Light Bulb Theorem.
problem Necessary conditions for the 4D Light Bulb Theorem.
method Constructing 4-manifolds with specific dual properties.
result Examples show the necessity of square zero assumption.
New theorem removes uniform finite upper bound for shrinkability of null decompositions.
problem Shrinkability of null decompositions with non-singleton elements.
method Defining squeezable and squashable subsets, proving their equivalence, and applying these definitions to null decompositions.
result Any null decomposition of a compact metric space whose non-singleton elements are recursively squeezable is shrinkable.