Brenda Johnson

Deriving calculus with cotriples

We construct a Taylor tower for functors from pointed categories to abelian categories via cotriples associated to cross effect functors. The tower was inspired by Goodwillies Taylor tower for functors of spaces, and is related to Dold and Puppes stable derived functors and Mac Lanes Q-construction. We study the layers, D n F = fiber(P n F → P n-1 F), and the limit of the tower. For the latter we determine a condition on the cross effects that guarantees convergence. We define differentials for functors, and establish chain and product rules for them. We conclude by studying exponential functors in this setting and describing their Taylor towers.

The derivatives of homotopy theory

We construct a functor of spaces, Mn , and show that its multilinearization is equivalent to the nth layer of the Taylor tower of the identity functor of spaces. This allows us to identify the derivatives of the identity functor and determine their homotopy type. The calculus of homotopy functors, developed by Goodwillie ([GI], [G2], [G3]), establishes that a homotopy functor, F, satisfying certain connectivity conditions, has associated to it a tower of functors, ... PF Pn-F .... These functors act like a Taylor series approximation to F in the sense that for a space, X, there is a map, PnF(X): F(X) -k PnF(X), for each n, and the connectivity of this map increases with n. This theory has been applied to the study of the functor A, Waldhausens algebraic K-theory of spaces. In this paper, we turn our attention to the Taylor tower of the identity functor of spaces, I. The ultimate goal is to identify the Taylor tower of I and use it to study homotopy theory. In this paper, we construct a collection of symmetric functors, {Mn}, and show that the multilinearization of Mn is equivalent to the nth layer, fiber (PnI Pn_ I), of the Taylor tower of I. This construction also allows us to identify the nth derivative of I. This is a spectrum with L,,action which is equivalent to the functor fiber (PnI -k PnI) . The paper is organized as follows. In section 1 we summarize the basic results and terminology of calculus that will be used throughout the paper. In section 2 we describe the problem of finding the Taylor tower of I in more detail and state the main results. In section 3 we outline the method by which the nth derivative of a functor is determined. In section 4 we construct the functor Mn and a natural transformation, Tn , used to establish the equivalence between the multilinearization of Mn and fiber(PnF PnF) . In section 5 we determine the homotopy type of the derivatives, and in section 6 we show that T. is sufficiertly connected to induce an equivalence between the multilinearization of Mn and fiber(P,F -PIF)l. The results in this paper come from the authors thesis, written uncler the direction of Tom Goodwillie at Brown University. The author wishes to thank him for his guidance and for many insightful discussions. The author alsQ wishes to thank Randy McCarthy for his helpful suggestions during the writing of this paper. Received by the editors February 10, 1994. 1991 Mathematics Subject Clqssification. Primary 55P65. ? 1995 American Mathematical Society 0002-9947/95

LINEARIZATION, DOLD-PUPPE STABILIZATION, AND MAC LANE'S Q-CONSTRUCTION

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Cross effects and calculus in an unbased setting

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Taylor towers for functors of additive categories

In this paper we study linear functors, i.e., functors of chain complexes of modules which preserve direct sums up to quasi-isomorphism, in order to lay the foundation for a further study of the Goodwillie calculus in this setting. We compare the methods of Dold and Puppe, Mac Lane, and Goodwillie for producing linear approximations to functors, and establish conditions under which these methods are equivalent. In addition, we classify linear functors in terms of modules over an explicit differential graded algebra. Several classical results involving Dold-Puppe stabilization and Mac Lane’s Q-construction are extended or given new proofs.

ON THE SIZE OF LINKS IN Kn, n, Kn, n, 1, AND Kn

We study functors F from C_f to D where C and D are simplicial model categories and C_f is the full subcategory of C consisting of objects that factor a fixed morphism f from A to B. We define the analogs of Eilenberg and Mac Lanes cross effects functors in this context, and identify explicit adjoint pairs of functors whose associated cotriples are the diagonals of the cross effects. With this, we generalize the cotriple Taylor tower construction of [10] from the setting of functors from pointed categories to abelian categories to that of functors from C_f to D to produce a tower of functors whose n-th term is a degree n functor. We compare this tower to Goodwillies tower of n-excisive approximations to F found in [8]. When D is a good category of spectra, and F is a functor that commutes with realizations, the towers agree. More generally, for functors that do not commute with realizations, we show that the terms of the towers agree when evaluated at the initial object of C_f.

Fixed Points and Fermat: A Dynamical Systems Approach to Number Theory

Given a functor F: A → B of additive categories, we construct a tower of functors … → PnF → Pn − 1F → Pn − 2F → … → P1F → F(0). We show that each PnF is degree n up to chain homotopy and, under certain assumptions, approximates F in a range that grows with n. We compare our Taylor tower with Goodwillies Taylor tower for a functor of spaces and establish conditions under which they are equivalent. This is a continuation of work by Johnson and McCarthy (to appear).

Women in Topology

For all n ≥ 4, and all even 4 ≤ s, t ≤ 2n - 4 with s + t = 2n, we show that every spatial embedding of Kn, n contains a non-trivial link involving and s-cycle and a t-cycle. We use this result to prove that for all n ≥ 3, 3 ≤ u, v ≤ 2n - 2 with u + v = 2n + 1, every spatial embedding of Kn, n, 1 has a non-trivial link involving a u-cycle and a v-cycle. These results imply that every embedding of Kn, for n ≥ 4, has non-trivial links of all possible sizes.

An Alpine Expedition through Algebraic Topology

For every prime p and all positive integers a, aP a (mod p). There are many proofs of this result; see [2] for some of them. It and related number-theoretic results are often used to establish facts about periodic points in dynamical systems [1, p. 119]. Our goal is to show at an elementary level how this process can be reversed: we use fixed and periodic point arguments to prove number-theoretic facts, including Fermats Little Theorem. The idea of obtaining number theoretic results via dynamical systems is not new. For instance, Furstenberg has shown the arithmetic progression theorems of van der Waerden and of Szemeredi can be derived from generalizations of the recurrence theorems of Birkhoff and of Poincare [4]. The results we present here are of a much more elementary nature. Our new proof of Fermats Little Theorem involves analyzing the fixed and periodic points of the following functions ga. For each integer a ? 2, let ga : [0, 1] [0, 1] be given by ~~~~~~~~1 la x for 0 < x

INTRINSICALLY KNOTTED GRAPHS HAVE AT LEAST 21 EDGES

We consider the action of p-toral subgroups of U(n) on the unitary partition complex Ln. We show that if H ⊆ U(n) has noncontractible fixed points on Ln, then the image of H in the projective unitary group U(n)/S is an elementary abelian pgroup.