# Mathematics Colloquia and Seminars

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### Generalized Property R and trisections of the 4-sphere

**Geometry/Topology**

Speaker: | Alex Zupan, Univ of Nebraska |

Location: | 2112 MSB |

Start time: | Tue, May 10 2016, 1:10PM |

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 decompositions of the standard smooth 4-sphere which require additional pairs of canceling handles before they admit simplification. For some decompositions, this problem reduces to a 3-dimensional conjecture known as the Generalized Property R Conjecture (GPRC). We can use trisections in this setting for a two-fold purpose: We resolve a weakening of the GPRC in certain cases, and we also give a new perspective on prominent potential counterexamples to the GPRC, describing connections to potential well-known counterexamples to the Andrews-Curtis Conjecture.