Homotopical Methods in Manifold Theory

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The PI will work on diverse problems in the algebraic topology of manifolds. The area of investigations are equivariant homotopical intersection theory, higher Reidemeister torsion, the construction of periodic families of Poincare duality spaces, and multiple disjunction problems for spaces of smooth embeddings. The methods to be employed on each of these projects have a common thread, involving equivariant and fiberwise homotopy theory. The proposed activity will strengthen collaborative mathematics between individuals at 5 universities (Brown, Notre Dame, Buffalo, Altoona and Wayne State). It will also foster the training of graduate students in algebraic topology at Wayne State.A "manifold" is a topological space that satisfies a homogeneity property. Locally speaking, all manifolds are alike in that at any point one sees a copy of Euclidean space. It is the global structure of manifolds that makes them interesting objects of study. Typically, algebraic topologists study manifolds by assigning certain algebraic quantities, called "invariants," to them, which measure their global topological structure. Manifolds having different invariants can then be distinguished from one another. Manifolds arise naturally in physics, chemistry and biology as spaces of solutions of a suitably "nice" set of algebraic equations modeling the scientific object of study (space-time, atoms, dynamical systems, etc.) . Manifolds play a central role in mathematics. It is often the case that mathematical questions about manifolds can be formulated in terms of parametrized families of functions between spaces associated with manifolds. Homotopy theory is a subject designed to tackle questions about such families of functions. The PI proposes to study certain kinds of manifold questions which can be analyzed from the homotopy theoretic toolbox.