Manifolds and calculus of functors

Brown University, Providence RI

Investigators

Abstract

Abstract Award: DMS-0708601 Principal Investigator: Thomas G. Goodwillie All of the proposed research is connected with the calculus of functors in its various forms. First, the main results of homotopy calculus will be extended to a natural and high degree of generality. Geometric language will be used to develop the analogy with ordinary differential calculus, so that a suitable homotopical category gets something like tangent and cotangent spaces, spaces of tensor fields, connections, differential operators, and so on. The language should be useful both for clarifying the ideas and for suggesting new directions. Second, the derivative of algebraic K-theory will be investigated in a general setting. Third, the part of manifold theory which in some sense comes next after stable pseudoisotopy (it could be called metastable pseudoisotopy) will be explored from the calculus viewpoint. Its derivatives will be studied by methods analogous to the Hochschild and cyclic homology methods that were relevant to the stable theory; where loops in a manifold were involved, now there should be maps of rank two graphs into the manifold. In addition to working out the metastable analogue of cyclic homology, there is the more fundamental question of seeking to understand the metastable phenomena in non-manifold terms as Waldhausen K-theory does for the stable theory, and calculus may hold clues to this. There is a sequence: (0) homotopy theory, where calculus leads to trees, (1) algebraic K-theory, where calculus leads to circles, and (2) metastable theory, where calculus leads to rank two graphs. One thing to be worked out is the relevant kinds of interactions between all these kinds of graphs in a manifold. Functor calculus involves extremes of abstraction, but it has down-to-earth roots. It is an organizing principle named for resemblance to the ordinary calculus of Newton and Leibniz. Sometimes a fact about numbers is best proved by placing it in a context where the number is part of a huge family of numbers -- a numerical function. Properties of the function can lead, by general theorems of calculus that at first seem like magic, to a computation of the number. The story here is similar: sometimes a fact about some mathematical entity -- not a number now, but perhaps a geometric object of some kind -- is best proved by placing it in a context where the object is part of a huge family of such objects -- a functor -- and using some magic of more recent vintage. One area of application of functor calculus is manifold topology. Another is homotopy theory. A manifold is a kind of mathematical system involving many variables. These are ubiquitous in science and mathematics. Manifold topology studies such systems for their own sake, treating them as geometric objects; an N-variable system is viewed as an N-dimensional object. Homotopy theory studies manifolds and other objects from a point of view in which a great deal of information is ignored, leaving only the coarsest features to consider. This change in viewpoint is a powerful idea, because this process of distillation leads to conceptual clarity and new methods. Besides being a key tool in the study of manifolds, homotopy theory is a branch of topology in its own right. Thus geometry grows, surprisingly and naturally, as the years go by. In functor calculus it becomes possible in a sense to view all of homotopy theory as a geometric object.