Functorial Calculus and Manifolds

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9806981 Goodwillie This project is mainly concerned with the long-range goal of investigating those aspects of the topology of manifolds that lie beyond the reach of algebraic K-theory. The method is a combination of several kinds of "functor calculus." One task is to investigate some interrelationships between these calculi with an eye to strengthening the general method. This is expected to involve some foundational work on Poincare duality complexes. Another task for the long haul is to look at what may be called a second-order analogue of algebraic K-theory; it arises by a kind of quadratic stabilization of manifold phenomena. For the short haul there is the job of constructing a kind of second-order analogue of cyclic homology that should be related to the above as cyclic homology is related to algebraic K-theory. There is at least a slight hope that this would yield topologically defined invariants for four-manifolds that are akin to the well-known analytically defined Seiberg-Witten invariants. The word "calculus" is used because of an attractive analogy with the ordinary calculus of functions. Sometimes, of course, the way to settle a numerical question is to see the answer as a value of a function; by means of the powerful apparatus of calculus, properties of the function can lead to a computation of the number. Likewise, a fact about some geometrically defined object is sometimes best proved by viewing the object as a particular value of a functor and using the apparatus of some kind of functor calculus. This analogy may show something of the flavor of the work; the content is harder to describe to the non-expert, because most of the "geometric" objects in question are connected to everyday reality by rather long chains of abstract ideas. ***