Celebratio Mathematica — Scharlemann

Complete Bibliography

Works connected to Denis Vogel

Filter the Bibliography List

year	title	people

			clear

M. Scharlemann: “An overview of Property 2R,” pp. 317–325 in The mathematics of knots. Edited by M. Banagl and D. Vogel. Contributions in Mathematical and Computational Sciences 1. Springer (Berlin), 2011. MR 2777854 Zbl 1221.57012 incollection

The celebrated Property R Conjecture, affirmed by David Gabai (J. Differ. Geom. 26:461, 1987), can be viewed as the first stage of a sequence of conjectures culminating in what has been called the Generalized Property R Conjecture. This conjecture is relevant to the study of outstanding problems in both 3-manifolds (specifically, links in \( S^3 \)) and 4-manifolds (specifically, the Schoenflies Conjecture and the smooth Poincaré Conjecture). Here we give an overview of part of forthcoming work of R. Gompf, A. Thompson and the author which considers the next stage in such a progression, called the Property 2R Conjecture.

It is shown that the lowest genus counterexample (if any exists) cannot be fibered. Exploiting Andrews–Curtis type considerations on presentations of the trivial group, it is argued that one of the simplest possible candidates for a counterexample, the square knot, probably is one. This suggests there is a genus one counterexample, though we have so far been unable to identify it. Finally, we note that the counterexample need not be an obstacle to the sort of 4-manifold consequences towards which the Generalized Property R Conjecture is aimed.

Denis Vogel	Related
Markus Banagl	Related