Mathematics Colloquia and Seminars

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  An n-component link is called an R-link if it has a surgery which yields an n-connected sum of S^1\times S^2 . An R-link is said to have Generalized Property R if it is handle-slide equivalent to the unlink. Problem 1.82 of Kirby’s Problem List conjectures that every R-link has Generalized Property R. The significance of the Generalized Property R conjecture is that it implies all geometrically simply-connected homotopy 4-sphere’s are diffeomorphic to the standard 4-sphere in a somewhat controlled way. In this talk, we extend the notion of Generalized Property R into the CP^2 setting by defining the notion Property S. With this, we also extend results of Gompf, Scharlemann, and Thompson on fibered knots into this setting as well.