Quantum Topology Seminar
Scott Carter
Univ of South Alabama
Folding branched coverings of the 3-sphere branched along a knot or link.
Abstract: A simple branched cover is one for which the branched points are simple. Locally they resemble the graph of the complex squaring function z \mapsto z^2. That is, in a closed tubular neighborhood of the knot, the squaring function occurs in a meridional direction.
Cyclic branched covers assign a permutation cycle to each of the arcs in the link diagram. For example, the Poincare homology sphere (PHS) is a 2-fold cyclic branched cover of the 3-sphere branched over the torus knot T(3,5). This is also a simple branched cover. The PHS is also a 3-fold cyclic branched cover of the 3-sphere branched along the torus knot T(2,5), or a 5-fold cyclic branched cover of the 3-sphere branched along the trefoil.
There are also branched covers that are associated to non-trivial homomorphisms from the fundamental group into a finite group.
Suppose that M denotes a branched cover of the 3-sphere that is branched along a knot or link. The theorem that I intend to prove is that there is a (mostly general position map) from M into S^3 \times [0,n+1] such that the composition of this map with the projection onto the sphere factor induces the branched covering. Such a map is called a {\it folding of the branched cover} M.
Explicit descriptions of the foldings of the PHS will be demonstrated, and other esoteric examples will be presented.
Thursday October 10, 2024 at 12:00 PM in Zoom