Heegaard diagrams and homotopy 3-spheres
This paper suggests a procedure for constructing homotopy 3-spheres via a computer search. In order to state the main result used in this procedure, let \(H_{r+s}\) denote a genus \(r+s\), 3-dimensional handlebody, which is assumed to be standardly embedded in \({\mathbb{R}}^ 3\). The collection of embedded curves \(\{\alpha_ 1,...,\alpha_ r,\beta_ 1,...,\beta_ s\}\subset \partial H_{r+s}\) is called an H-system provided (i) this collection does not separate \(\partial H_{r+s}\), (ii) there exist disjoint, compact, orientable surfaces \(F_ 1,...,F_ r\) in \({\mathbb{R}}^ 3-int H_{r+s}\) such that \(\partial F_ i=\alpha_ i\), and (iii) there exist disjointly embedded 2-disks \(D_ 1,...,D_ s\) in \(H_{r+s}\) such that \(\partial D_ i=\beta_ i.\) Theorem: Corresponding to each H-system is a homotopy 3-sphere. Moreover, every homotopy 3-sphere occurs in this way. Briefly, if \(\{\alpha_ 1,...,\alpha_ r,\beta_ 1,...,\beta_ s\}\) is an H-system, by cutting \(H_{r+s}\) along the \(D_ i's\) and adding 2-handles along the \(\alpha_ i's\), one obtains a homotopy 3-cell. Simple algorithms are given for determining when a collection of curves \(\{\alpha_ i,...,\alpha_ r,\beta_ 1,...,\beta_ s\}\subset \partial H_{r+s}\) satisfy conditions (ii) and (iii) - thereby making a computer search viable.
homotopy 3-spheres via a computer search
scientific article; zbMATH DE number 4054612