Calculus of Functors and Applications

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DMS-0307069Gregory Z. AroneThe main goal of this project is to apply calculus of functors, especially the"orthogonal calculus" version developed by M. Weiss, to the study of spaces ofembeddings. In more detail, let M, N be smooth manifolds. The PI would like tostudy the space of embeddings Emb(M,N) by taking the cartesian product of one ortwo of these manifolds with a generic Euclidean space, and investigating theproperties of the obtained functor of the Eucliedan space. The general theory ofcalculus associates with such a functor a sequence of "derivatives", where then-th derivative is a spectrum with an action of the orthogonal group O(n), and a"Taylor tower" - a sequence of approximations by polynomial functors. The studyof the derivatives of the embedding functor leads one to consider a ratherbeautiful array of topological constructions, some classical, some new. Theclassical constructions that one encounters include (a generalization of) theFulton-McPherson compactification, partition posets and spaces of trees. The PIhopes that this project will yield significant new insights into the topology ofspaces of embeddings and automorphisms of manifolds.Manifolds are among the basic objects of study in mathematics. Manifolds come indifferent dimensions. One dimensional manifolds are curves, two dimensionalmanifolds are surfaces, and high dimensional manifolds are suitable extensionsof these concepts. One of the basic questions about manifolds is: given amanifold M, what are the possible symmetries (diffeomorphisms) of M? It was T.Goodwillie who came up with the very striking idea that rather than approachsuch questions "one manifold at at a time", one should study systematically howthe space of symmetries (or whatever it is we want to study) changes, as onevaries the manifold. This results in a theory analogous to the classicaldifferential calculus, where functions are studied via their derivatives, Taylorpolynomials and so forth. This idea provides one with a powerful and beautiful framework for studying manifolds (and other objects of interest in mathematics, especially topology), subsumes a fair amount of classical techniques, and leads one to discover beautiful new constructions in topology.