Open Collections

Description

Waldhausen's Theorem implies that any handle decomposition of the 3­-sphere can be simplified without introducing additional handles. The analogue in dimension 4 is unknown, but it is widely believed that there are handle decompositions of the standard smooth 4-­sphere which require additional pairs of canceling handles before they admit simplification. For handle decompositions without 1­-handles, it suffices to understand links in the 3-­sphere with certain Dehn surgeries ­­ these are the links characterized by the Generalized Property R Conjecture and its variations. We show that a given link has Stable Generalized Property R if and only if a certain infinite family of induced trisections is non­standard; thus, we provide evidence that trisections may be used to verify that the potential counterexamples introduced by Gompf-Scharlemann­-Thompson do not have Stable Generalized Property R. Parts of this talk are joint with Jeffrey Meier and Trenton Schirmer.