• Corpus ID: 219558930

The 4-dimensional disc embedding theorem and dual spheres

  title={The 4-dimensional disc embedding theorem and dual spheres},
  author={Mark Powell and Arunima Ray and Peter Teichner},
  journal={arXiv: Geometric Topology},
We modify the proof of the disc embedding theorem for 4-manifolds, which appeared as Theorem 5.1A in the book "Topology of 4-manifolds" by Freedman and Quinn, in order to construct geometrically dual spheres. These were claimed in the statement but not constructed in the proof. Fundamental results in 4-manifold topology such as the existence and exactness of the surgery sequence, the s-cobordism theorem, and thence the classification of closed, simply connected topological 4-manifolds up to… 
7 Citations
The relative Whitney trick and its applications
We introduce a geometric operation, which we call the relative Whitney trick, that removes a single double point between properly immersed surfaces in a 4-manifold with boundary. Using the relative
Introduction to Whitney Towers
These introductory notes on Whitney towers in 4-manifolds, as developed in collaboration with Jim Conant and Peter Teichner, are an expansion of three expository lectures given at the Winter Braids X
Counterexamples in 4-manifold topology
. We consider several of the most commonly studied equivalence relations on 4-manifolds and how they are related to one another. We explain implications e.g. that h -cobordant manifolds are stably
Embedding surfaces in 4-manifolds
We prove a surface embedding theorem for 4-manifolds with good fundamental group in the presence of dual spheres, with no restriction on the normal bundles. The new obstruction is a Kervaire–Milnor
Homotopy versus isotopy: Spheres with duals in 4-manifolds
David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in the absence of 2-torsion in the fundamental group. We extend his result to 4-manifolds with arbitrary fundamental group by
Minimal Euler Characteristics for Even-Dimensional Manifolds with Finite Fundamental Group
We consider the Euler characteristics χ(M) of closed orientable topological 2n–manifolds with (n−1)–connected universal cover and a given fundamental group G of type Fn. We define q2n(G), a
On non‐orientable surfaces embedded in 4‐manifolds
We find conditions under which a non‐orientable closed surface smoothly embedded into an orientable 4 ‐manifold X can be represented by a connected sum of an embedded closed surface in X and an


4-Manifold topology I: Subexponential groups
The technical lemma underlying the 5-dimensional topologicals-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating π1-null
Isotopy of 4-manifolds
The principal result of this paper is that the group of homeomorphisms mod isotopy (the "homeotopy" group) of a closed simply-connected 4-manifold is equal to the automorphism group of the quadratic
In this paper we introduce a method for the investigation of smooth simply connected manifolds of dimension n ≥ 5 that permits a classification of them with exactness up to orientation-preserving
Closed oriented 4-manifolds with the same geometrically two-dimensional fundamental group (satisfying certain properties) are classified up to s-cobordism by their w2-type, equivariant intersection
On the Structure of Manifolds
In this paper, we prove a number of theorems which give some insight into the structure of differentiable manifolds. The methods, results and some notation of [13], hereafter referred to as GPC, and
Triangulating and Smoothing Homotopy Equivalences and Homeomorphisms. Geometric Topology Seminar Notes
We will study the smooth and piecewise linear manifolds within a given homotopy equivalence class. In the first part we find an obstruction theory for deforming a homotopy equivalence between
Surgery and duality
Surgery, as developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types of
The topology of four-dimensional manifolds
0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's. Of the mysteries still remaining after that period of great success the most compelling seemed to lie in
Stable mappings and their singularities
I: Preliminaries on Manifolds.- 1. Manifolds.- 2. Differentiable Mappings and Submanifolds.- 3. Tangent Spaces.- 4. Partitions of Unity.- 5. Vector Bundles.- 6. Integration of Vector Fields.- II:
Surgery on compact manifolds
Preliminaries: Note on conventions Basic homotopy notions Surgery below the middle dimension Appendix: Applications Simple Poincare complexes The main theorem: Statement of results An important