Geometric Classification Theory and Invariants of 4-Manifolds

Classification of four-dimensional shapes, known in mathematics as manifolds, is a fundamental problem in geometry with deep connections to physics. An approach to studying manifolds called surgery theory, which has proved to be very fruitful in higher dimensions, involves cutting and re-assembling them from smaller parts which are easier to understand. A breakthrough in the 1980s showed that this theory holds for a certain type of four-dimensional manifolds. One of the goals of this project is to examine its validity in full generality. The project also aims to study symmetries of four-dimensional spaces, and to develop new tools for distinguishing them. Broader impacts of the project include research opportunities for undergraduate students, mentoring graduate students and postdoctoral fellows, and outreach activities for middle school students. The PI will consider an approach to the 4-dimensional topological surgery conjecture for free non-abelian fundamental groups using a novel framework for the embedding problem of a collection of 4-dimensional 2-handlebodies in 4-space. The project will investigate properties of an invariant defined using the notion of a derived link of a handlebody. The PI will also consider another aspect of geometric classification theory, using pseudo-isotopy methods to study mapping class groups of both topological and smooth 4-manifolds. Another focus of the project is to study topological applications of a collection of invariants developed by the PI, based on ideas in stable homotopy theory and in link homology. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.