Manifolds and calculus of functors

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AbstractAward: DMS-0708601Principal Investigator: Thomas G. GoodwillieAll of the proposed research is connected with the calculus offunctors in its various forms. First, the main results ofhomotopy calculus will be extended to a natural and high degreeof generality. Geometric language will be used to develop theanalogy with ordinary differential calculus, so that a suitablehomotopical category gets something like tangent and cotangentspaces, spaces of tensor fields, connections, differentialoperators, and so on. The language should be useful both forclarifying the ideas and for suggesting new directions. Second,the derivative of algebraic K-theory will be investigated in ageneral setting. Third, the part of manifold theory which in somesense comes next after stable pseudoisotopy (it could be calledmetastable pseudoisotopy) will be explored from the calculusviewpoint. Its derivatives will be studied by methods analogousto the Hochschild and cyclic homology methods that were relevantto the stable theory; where loops in a manifold were involved,now there should be maps of rank two graphs into the manifold. Inaddition to working out the metastable analogue of cyclichomology, there is the more fundamental question of seeking tounderstand the metastable phenomena in non-manifold terms asWaldhausen K-theory does for the stable theory, and calculus mayhold clues to this. There is a sequence: (0) homotopy theory,where calculus leads to trees, (1) algebraic K-theory, wherecalculus leads to circles, and (2) metastable theory, wherecalculus leads to rank two graphs. One thing to be worked out isthe relevant kinds of interactions between all these kinds ofgraphs in a manifold.Functor calculus involves extremes of abstraction, but it hasdown-to-earth roots. It is an organizing principle named forresemblance to the ordinary calculus of Newton andLeibniz. Sometimes a fact about numbers is best proved by placingit in a context where the number is part of a huge family ofnumbers -- a numerical function. Properties of the function canlead, by general theorems of calculus that at first seem likemagic, to a computation of the number. The story here is similar:sometimes a fact about some mathematical entity -- not a numbernow, but perhaps a geometric object of some kind -- is bestproved by placing it in a context where the object is part of ahuge family of such objects -- a functor -- and using some magicof more recent vintage. One area of application of functorcalculus is manifold topology. Another is homotopy theory. Amanifold is a kind of mathematical system involving manyvariables. These are ubiquitous in science andmathematics. Manifold topology studies such systems for their ownsake, treating them as geometric objects; an N-variable system isviewed as an N-dimensional object. Homotopy theory studiesmanifolds and other objects from a point of view in which a greatdeal of information is ignored, leaving only the coarsestfeatures to consider. This change in viewpoint is a powerfulidea, because this process of distillation leads to conceptualclarity and new methods. Besides being a key tool in the studyof manifolds, homotopy theory is a branch of topology in its ownright. Thus geometry grows, surprisingly and naturally, as theyears go by. In functor calculus it becomes possible in a senseto view all of homotopy theory as a geometric object.