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Juanita Pinzon Caicedo, NC State, Four–manifolds and knot concordance
November 29, 2018 | 2:00 pm - 3:00 pm EST
The main goal of geometric topology is the classification of manifolds within a certain framework (topological, piecewise linear, smooth, simply-connected, symplectic, etc.). Dimension four is special, as it is the only dimension in which a manifold can admit infinitely many non-equivalent smooth structures, and the only dimension in which there exist manifolds homeomorphic but not diffeomorphic to R4. In turn, knot concordance is the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Questions pertaining to 4—manifolds, like the difference between topological and smooth structures, can be addressed in terms of knot concordance. A powerful tool for studying the algebraic structure of the knot concordance group comes from satellite operations. In the talk I will describe how to use SO(3) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group.