9703655 McCarthy Tom Goodwillie defined a Taylor tower for homotopy functors. These are functors from spaces (or spectra) to spaces (or spectra) and they must preserve homotopy equivalences. The tower is an inverse limit of functors with the fibers between stages being equivariant homology theories. The tower does not necessarily recover the original functor for all spaces or spectra but does for suitably connected objects. One way in which the investigator will study this tower is by studying another tower that agrees with the Taylor tower within the radius of convergence for a given functor but whose defining characteristic at the n-th level is the universal degree n construction as compared to the universal n-excisive construction given by the Taylor tower. He will also continue to study algebraic K-theory by use of linearizing functors of exact categories to spectra. A functor F from exact categories to spaces or spectra is said to satisfy ``additivity'' if F applied to the exact category of short exact sequences of a category is naturally equivalent to the two-fold product of F applied to the category given by the exact functors that take kernel and cokernel of a short exact sequence. There is a universal construction, called ``linearizing,'' which takes an arbitrary functor from exact categories to spectra and produces a new functor of exact categories that satisfies additivity. This is completely analogous to taking the ``derivative'' in the sense of Goodwillie when reinterpreted with these new towers. Algebraic K-theory itself is the linearization of a free functor from exact categories to spectra. Topological Hochschild homology and topological cyclic homology are also examples of linearizing a functor, and the trace maps that connect these theories may be studied from this point of view. The investigator will be examining different constructions for the study of algebraic K-theory obtained by linearizing various types of functors. The hope is to obtain new theories that one can use to study algebraic K-theory effectively. An important tool for studying functions from the complex numbers to itself is the Taylor series expansion of the function about a point. For a functor -- a kind of generalized function -- from spaces to spaces, Tom Goodwillie has similarly defined a Taylor series expansion of the functor about a space. In standard analysis one must assume a function has all its derivatives about a point to ensure that the Taylor tower exists, and then one can be sure that this approximation to the original function is accurate only within a radius of convergence about the point. Similarly, for functors of spaces, one must make assumptions about the functor to ensure its Taylor tower exists, and even when these are satisfied, one obtains accurate estimates of the original functor only for spaces sufficiently close to the space of expansion. The investigator will be examining a modification of Goodwillie's original definition for the Taylor tower of functors from spaces to spaces that tends to agree with his definition for spaces within the radius of convergence of a point but differs in general. Two advantages of this new tower are that it is easier to define and that it can be applied to an even greater variety of interesting situations. One area to be explored with this new technology is algebraic K-theory, which from this point of view is simply the derivative of a particularly easy functor. ***
