Algebraic K-Theory in Fixed-Point Theory and Smooth Manifolds

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Fixed-point theory studies the fixed or stationary states in dynamical systems, which are critical to questions in topology, analysis, and even economics. Smooth manifolds are a special class of high-dimensional shapes that are of fundamental importance in geometry and physics. This project is concerned with important and unexpected connections between these two fields. To be more precise, they can be related using algebraic K-theory, a deep and abstract but pervasive subject whose reach already encompasses several mathematical disciplines, including homotopy theory, differential topology, algebraic geometry, and number theory. These research projects will develop these connections, giving us new ways to think about classical problems and to use techniques from fixed-point theory in unexpected places. Broader impacts include mentorship, workshop organization, and the writing of foundational works to lower the entry barrier into these disciplines.Four groups of projects will be pursued. The first shows that the Fuller trace, a certain noncommutative trace in genuine G-spectra, provides a complete invariant for removing n-periodic points from a map by a homotopy. The second uses the trace from K-theory of endomorphisms to topological restriction homology (TR) to show that this Fuller trace is a topological, non-commutative generalization of the characteristic polynomial from linear algebra. The third project proves the equivariant refinement of Waldhausen's parametrized h-cobordism theorem, a fundamental and celebrated result in high-dimensional manifold theory. The fourth project uses noncommutative traces to compute transfer maps in algebraic K-theory, leading to torsion calculations that identify exotic bundles and fibrations that cannot be stably smoothed. In other words, fixed-point theory techniques can be used to understand smooth structures.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.